Jungfraujoch/image_analysis/lattice_search/LePageLattice.cpp

// SPDX-FileCopyrightText: 2026 Filip Leonarski, Paul Scherrer Institute <filip.leonarski@psi.ch>
// SPDX-License-Identifier: GPL-3.0-only

// Bravais lattice from the metric symmetry OPERATORS, rather than from a table of reduced-cell
// characters. The two-fold search is Le Page's (1982) J. Appl. Cryst. 15, 255-259, vendored in
// gemmi (twin.hpp); what is here is the step after it - turning the rotation group it finds into
// a conventional cell, a centring letter and an integral change of basis.
//
// Following Le Page (1982) J. Appl. Cryst. 15, 255-259 and
// Grosse-Kunstleve (1999) Acta Cryst. A55, 383-395

#include "LePageLattice.h"
#include "../../common/JFJochMath.h"

#include <gemmi/cellred.hpp>
#include <gemmi/twin.hpp>

#include <algorithm>
#include <array>
#include <cmath>
#include <numeric>

namespace {

using Mat3i = std::array<int, 9>;               // row-major
using Vec3i = std::array<int, 3>;

constexpr Mat3i kIdentity3 = {1, 0, 0, 0, 1, 0, 0, 0, 1};

int At(const Mat3i &m, int r, int c) { return m[3 * r + c]; }

Mat3i MatMul(const Mat3i &a, const Mat3i &b) {
    Mat3i r{};
    for (int i = 0; i < 3; i++)
        for (int j = 0; j < 3; j++) {
            int s = 0;
            for (int k = 0; k < 3; k++)
                s += At(a, i, k) * At(b, k, j);
            r[3 * i + j] = s;
        }
    return r;
}

int Det3(const Mat3i &m) {
    return At(m,0,0) * (At(m,1,1)*At(m,2,2) - At(m,1,2)*At(m,2,1))
         - At(m,0,1) * (At(m,1,0)*At(m,2,2) - At(m,1,2)*At(m,2,0))
         + At(m,0,2) * (At(m,1,0)*At(m,2,1) - At(m,1,1)*At(m,2,0));
}

// adj(M), so that M^-1 = adj(M) / det(M) - kept integral to determine the centring exactly.
Mat3i Adjugate(const Mat3i &m) {
    Mat3i a{};
    a[0] =  (At(m,1,1)*At(m,2,2) - At(m,1,2)*At(m,2,1));
    a[1] = -(At(m,0,1)*At(m,2,2) - At(m,0,2)*At(m,2,1));
    a[2] =  (At(m,0,1)*At(m,1,2) - At(m,0,2)*At(m,1,1));
    a[3] = -(At(m,1,0)*At(m,2,2) - At(m,1,2)*At(m,2,0));
    a[4] =  (At(m,0,0)*At(m,2,2) - At(m,0,2)*At(m,2,0));
    a[5] = -(At(m,0,0)*At(m,1,2) - At(m,0,2)*At(m,1,0));
    a[6] =  (At(m,1,0)*At(m,2,1) - At(m,1,1)*At(m,2,0));
    a[7] = -(At(m,0,0)*At(m,2,1) - At(m,0,1)*At(m,2,0));
    a[8] =  (At(m,0,0)*At(m,1,1) - At(m,0,1)*At(m,1,0));
    return a;
}

Vec3i MulVec(const Mat3i &m, const Vec3i &v) {
    return {At(m,0,0)*v[0] + At(m,0,1)*v[1] + At(m,0,2)*v[2],
            At(m,1,0)*v[0] + At(m,1,1)*v[1] + At(m,1,2)*v[2],
            At(m,2,0)*v[0] + At(m,2,1)*v[1] + At(m,2,2)*v[2]};
}

Vec3i Cross(const Vec3i &a, const Vec3i &b) {
    return {a[1]*b[2] - a[2]*b[1], a[2]*b[0] - a[0]*b[2], a[0]*b[1] - a[1]*b[0]};
}

int Gcd3(const Vec3i &v) {
    int g = std::gcd(std::gcd(std::abs(v[0]), std::abs(v[1])), std::abs(v[2]));
    return g;
}

// The shortest lattice vector along a direction is the direction divided by the gcd of its
// components, and its sign is fixed so the same axis is always the same vector.
bool MakePrimitive(Vec3i &v) {
    const int g = Gcd3(v);
    if (g == 0)
        return false;
    for (int &x : v)
        x /= g;
    for (int i = 0; i < 3; i++) {
        if (v[i] > 0) break;
        if (v[i] < 0) { v[0] = -v[0]; v[1] = -v[1]; v[2] = -v[2]; break; }
    }
    return true;
}

int Trace(const Mat3i &m) { return m[0] + m[4] + m[8]; }

// A proper rotation has trace 1 + 2 cos(theta): 3, -1, 0, 1, 2 for orders 1, 2, 3, 4, 6.
int RotationOrder(const Mat3i &m) {
    switch (Trace(m)) {
        case  3: return 1;
        case -1: return 2;
        case  0: return 3;
        case  1: return 4;
        case  2: return 6;
        default: return 0;
    }
}

// The +1 eigenvector of a rotation of order >= 2: (R - I) has rank 2, so the cross product of any
// two independent of its rows spans the kernel. Exact in integers.
bool RotationAxis(const Mat3i &m, Vec3i &axis) {
    Mat3i d = m;
    d[0] -= 1; d[4] -= 1; d[8] -= 1;
    const Vec3i rows[3] = {{d[0], d[1], d[2]}, {d[3], d[4], d[5]}, {d[6], d[7], d[8]}};
    Vec3i best{0, 0, 0};
    long long best_norm = 0;
    for (int i = 0; i < 3; i++)
        for (int j = i + 1; j < 3; j++) {
            const Vec3i c = Cross(rows[i], rows[j]);
            const long long n = 1LL*c[0]*c[0] + 1LL*c[1]*c[1] + 1LL*c[2]*c[2];
            if (n > best_norm) { best_norm = n; best = c; }
        }
    if (best_norm == 0)
        return false;
    axis = best;
    return MakePrimitive(axis);
}

// Metric tensor of the reduced primitive cell, so integer lattice vectors can be measured.
struct Metric {
    double g[3][3];
    double Dot(const Vec3i &u, const Vec3i &v) const {
        double s = 0;
        for (int i = 0; i < 3; i++)
            for (int j = 0; j < 3; j++)
                s += g[i][j] * u[i] * v[j];
        return s;
    }
    double Len(const Vec3i &u) const { return std::sqrt(std::max(0.0, Dot(u, u))); }
};

Metric MetricOf(const gemmi::UnitCell &c) {
    Metric m{};
    const double ca = std::cos(gemmi::rad(c.alpha)), cb = std::cos(gemmi::rad(c.beta)),
                 cg = std::cos(gemmi::rad(c.gamma));
    m.g[0][0] = c.a * c.a;   m.g[1][1] = c.b * c.b;   m.g[2][2] = c.c * c.c;
    m.g[0][1] = m.g[1][0] = c.a * c.b * cg;
    m.g[0][2] = m.g[2][0] = c.a * c.c * cb;
    m.g[1][2] = m.g[2][1] = c.b * c.c * ca;
    return m;
}

// The rotation group generated by the two-folds Le Page found, built one generator at a time in
// order of increasing obliquity and only while the result is still a possible lattice rotation
// group. Two-folds accepted at a loose obliquity need not be consistent with each other - closing
// over an inconsistent set runs away - so a generator whose closure is not one of the seven orders
// a holohedry can have is dropped instead of aborting the whole search.
bool IsHolohedryOrder(size_t n) {
    return n == 1 || n == 2 || n == 4 || n == 6 || n == 8 || n == 12 || n == 24;
}

std::vector<Mat3i> CloseGroup(const std::vector<Mat3i> &gens) {
    std::vector<Mat3i> group{kIdentity3};
    for (const Mat3i &g : gens) {
        if (std::find(group.begin(), group.end(), g) != group.end())
            continue;
        std::vector<Mat3i> trial = group;
        trial.push_back(g);
        bool ok = true;
        for (size_t i = 0; i < trial.size() && ok; i++)
            for (size_t j = 0; j < trial.size() && ok; j++) {
                const Mat3i p = MatMul(trial[i], trial[j]);
                if (std::find(trial.begin(), trial.end(), p) == trial.end()) {
                    if (trial.size() >= 24) { ok = false; break; }
                    trial.push_back(p);
                }
            }
        if (ok && IsHolohedryOrder(trial.size()))
            group = std::move(trial);
    }
    return group;
}

// All integer vectors v with r . v = 0 form a plane lattice; a pair of them is a BASIS of it
// exactly when their cross product is +/- r (r primitive). Enumerating short v and taking the two
// shortest that pass that test gives the reduced basis of the plane - in two dimensions the
// successive minima are always reachable by a basis.
bool PlaneBasis(const Vec3i &normal, const Metric &metric, Vec3i &p, Vec3i &q) {
    Vec3i r = normal;
    if (!MakePrimitive(r))
        return false;
    for (int box = 4; box <= 24; box *= 2) {
        std::vector<Vec3i> cand;
        for (int i = -box; i <= box; i++)
            for (int j = -box; j <= box; j++)
                for (int k = -box; k <= box; k++) {
                    const Vec3i v{i, j, k};
                    if (v[0] == 0 && v[1] == 0 && v[2] == 0)
                        continue;
                    if (r[0]*i + r[1]*j + r[2]*k != 0)
                        continue;
                    cand.push_back(v);
                }
        std::sort(cand.begin(), cand.end(), [&](const Vec3i &a, const Vec3i &b) {
            return metric.Dot(a, a) < metric.Dot(b, b);
        });
        for (const Vec3i &v1 : cand) {
            for (const Vec3i &v2 : cand) {
                const Vec3i c = Cross(v1, v2);
                if ((c[0] == r[0] && c[1] == r[1] && c[2] == r[2]) ||
                    (c[0] == -r[0] && c[1] == -r[1] && c[2] == -r[2])) {
                    p = v1; q = v2;
                    return true;
                }
            }
            break;      // v1 is the shortest vector of the plane; only it can start a reduced basis
        }
    }
    return false;
}

// The centring translations of the primitive lattice inside the conventional cell M: the rows of
// M^-1 generate them, and |det M| of them exist. Kept in units of 1/det so the arithmetic is exact.
std::vector<std::array<int, 3>> CentringTranslations(const Mat3i &M, int &den) {
    den = std::abs(Det3(M));
    const Mat3i adj = Adjugate(M);
    const int sign = Det3(M) > 0 ? 1 : -1;
    std::vector<std::array<int, 3>> out;
    for (int i = 0; i < den; i++)
        for (int j = 0; j < den; j++)
            for (int k = 0; k < den; k++) {
                std::array<int, 3> t{};
                for (int c = 0; c < 3; c++) {
                    int s = sign * (i * At(adj, 0, c) + j * At(adj, 1, c) + k * At(adj, 2, c));
                    s %= den;
                    if (s < 0) s += den;
                    t[c] = s;
                }
                if (std::find(out.begin(), out.end(), t) == out.end())
                    out.push_back(t);
            }
    std::sort(out.begin(), out.end());
    return out;
}

// Name the centring from those translations. Anything not on this list is not a Bravais lattice
// and means the conventional axes were built wrongly.
char CentringSymbol(const std::vector<std::array<int, 3>> &t, int den) {
    auto has = [&](int x, int y, int z) {
        return std::find(t.begin(), t.end(), std::array<int, 3>{x, y, z}) != t.end();
    };
    if (den == 1)
        return 'P';
    if (den == 2) {
        if (has(1, 1, 0)) return 'C';
        if (has(0, 1, 1)) return 'A';
        if (has(1, 0, 1)) return 'B';
        if (has(1, 1, 1)) return 'I';
    }
    if (den == 3) {
        if (has(1, 2, 2)) return 'R';        // obverse
        if (has(2, 1, 2)) return 'r';        // reverse - caller converts
    }
    if (den == 4 && has(0, 2, 2) && has(2, 0, 2) && has(2, 2, 0))
        return 'F';
    return '?';
}

Mat3i Transpose(const Mat3i &m) {
    return {m[0], m[3], m[6], m[1], m[4], m[7], m[2], m[5], m[8]};
}

Mat3i FromRows(const Vec3i &a, const Vec3i &b, const Vec3i &c) {
    return {a[0], a[1], a[2], b[0], b[1], b[2], c[0], c[1], c[2]};
}

gemmi::Mat33 ToMat33(const Mat3i &m) {
    return gemmi::Mat33(m[0], m[1], m[2], m[3], m[4], m[5], m[6], m[7], m[8]);
}

}   // namespace

std::optional<LePageResult> LePageLattice(const CrystalLattice &L, double max_obliquity_deg,
                                          const LePageOperatorFilter &keep_operator) {
    const UnitCell uc = L.GetUnitCell();
    gemmi::UnitCell g_uc(uc.a, uc.b, uc.c, uc.alpha, uc.beta, uc.gamma);
    gemmi::GruberVector gv(g_uc, 'P', true);
    // Same scaled epsilon as the character table: the type decision is on scalar products of order
    // 10^3-10^5 A^2 carried in a float cell.
    gv.niggli_reduce(1e-5 * std::max({gv.A, gv.B, gv.C}));

    // The reduction's change of basis, C, as an integer matrix. CrystalLattice::Multiply combines
    // basis ROWS, so a_red[i] = sum_j C[i][j] a_L[j]; fractional coordinates then map x_L = C^T x_red
    // and Miller indices h_red = C h_L, and an operator's Miller-index matrix maps M_L = C^-1 M_red C.
    // A Niggli reduction is unimodular, so C^-1 = det(C) * adj(C) is integral too - which is what lets
    // the filter below be offered an exact integer matrix in the caller's own basis.
    Mat3i C = kIdentity3;
    if (gv.change_of_basis) {
        const gemmi::Mat33 c = gemmi::rot_as_mat33(gv.change_of_basis->rot).transpose();
        for (int i = 0; i < 3; i++)
            for (int j = 0; j < 3; j++) {
                C[3 * i + j] = static_cast<int>(std::lround(c[i][j]));
                if (std::fabs(c[i][j] - C[3 * i + j]) > 1e-6)
                    return std::nullopt;
            }
    }
    const int det_C = Det3(C);
    if (det_C != 1 && det_C != -1)
        return std::nullopt;
    Mat3i C_inv = Adjugate(C);
    for (int &x : C_inv)
        x *= det_C;

    CrystalLattice L_red = L;
    if (gv.change_of_basis)
        L_red = L.Multiply(gemmi::rot_as_mat33(gv.change_of_basis->rot).transpose());

    const gemmi::UnitCell reduced = gv.get_cell();
    const Metric metric = MetricOf(reduced);

    // Le Page: the 81 two-folds a reduced cell can carry, ranked by obliquity.
    const std::vector<gemmi::OpObliquity> two_folds = gemmi::find_lattice_2fold_ops(reduced, max_obliquity_deg);

    std::vector<Mat3i> gens;
    gens.reserve(two_folds.size());
    double worst_obliquity = 0;
    for (const auto &[op, delta] : two_folds) {
        Mat3i m{};
        bool ok = true;
        for (int i = 0; i < 3 && ok; i++)
            for (int j = 0; j < 3 && ok; j++) {
                const int v = op.rot[i][j];
                if (v % gemmi::Op::DEN != 0) ok = false;
                m[3 * i + j] = v / gemmi::Op::DEN;
            }
        if (!ok)
            continue;
        if (keep_operator) {
            // A two-fold is its own inverse, so the Miller-index matrix of the operation x -> R x is
            // R^T; carry it into the basis of L as M_L = C^-1 M_red C (see C above).
            const Mat3i m_L = MatMul(C_inv, MatMul(Transpose(m), C));
            if (!keep_operator(ToMat33(m_L)))
                continue;
        }
        gens.push_back(m);
    }

    const std::vector<Mat3i> group = CloseGroup(gens);

    // A filtered search may promote only on two-folds the filter kept. Closure can bring back one it
    // rejected - two two-folds sixty degrees apart generate a three-fold and with it the whole
    // hexagonal group - and a sub-lattice whose own two-folds were scored and refused is not one the
    // evidence supports. Refuse rather than hand back a group resting on a rejected operator.
    if (keep_operator)
        for (const Mat3i &g : group)
            if (RotationOrder(g) == 2 && std::find(gens.begin(), gens.end(), g) == gens.end())
                return std::nullopt;
    for (const auto &[op, delta] : two_folds) {
        Mat3i m{};
        for (int i = 0; i < 3; i++)
            for (int j = 0; j < 3; j++)
                m[3 * i + j] = op.rot[i][j] / gemmi::Op::DEN;
        if (std::find(group.begin(), group.end(), m) != group.end())
            worst_obliquity = std::max(worst_obliquity, delta);
    }

    // Sort the group's operators by rotation order, and record the axis of each.
    struct Axis { int order; Vec3i dir; Mat3i rot; };
    std::vector<Axis> axes;
    int n_order[7] = {0, 0, 0, 0, 0, 0, 0};
    for (const Mat3i &m : group) {
        const int order = RotationOrder(m);
        if (order == 0 || Det3(m) != 1)
            return std::nullopt;                 // not a proper rotation - refuse rather than guess
        n_order[order]++;
        if (order == 1)
            continue;
        Vec3i dir{};
        if (!RotationAxis(m, dir))
            return std::nullopt;
        axes.push_back({order, dir, m});
    }

    gemmi::CrystalSystem system;
    switch (group.size()) {
        case 1:  system = gemmi::CrystalSystem::Triclinic; break;
        case 2:  system = gemmi::CrystalSystem::Monoclinic; break;
        case 4:  system = gemmi::CrystalSystem::Orthorhombic; break;
        case 6:  system = gemmi::CrystalSystem::Trigonal; break;
        case 8:  system = gemmi::CrystalSystem::Tetragonal; break;
        case 12: system = n_order[6] > 0 ? gemmi::CrystalSystem::Hexagonal : gemmi::CrystalSystem::Cubic; break;
        case 24: system = gemmi::CrystalSystem::Cubic; break;
        default: return std::nullopt;
    }

    // Conventional axes. Every one of them is the SHORTEST lattice vector along a symmetry
    // direction, which is what makes them integral combinations of the reduced primitive basis and
    // the change of basis an integer matrix.
    auto shortest_along = [&](const Vec3i &d) { Vec3i v = d; MakePrimitive(v); return v; };

    // The two-folds perpendicular to the principal axis are exactly those that send it to minus
    // itself - an integer test, no angle tolerance.
    auto perpendicular_two_folds = [&](const Vec3i &principal) {
        std::vector<Vec3i> out;
        for (const Axis &ax : axes) {
            if (ax.order != 2)
                continue;
            const Vec3i w = MulVec(ax.rot, principal);
            if (w[0] == -principal[0] && w[1] == -principal[1] && w[2] == -principal[2])
                out.push_back(shortest_along(ax.dir));
        }
        std::sort(out.begin(), out.end(), [&](const Vec3i &a, const Vec3i &b) {
            return metric.Dot(a, a) < metric.Dot(b, b);
        });
        return out;
    };

    auto principal_of_order = [&](int order) -> std::optional<Axis> {
        for (const Axis &ax : axes)
            if (ax.order == order)
                return ax;
        return std::nullopt;
    };

    Mat3i M = kIdentity3;

    switch (system) {
        case gemmi::CrystalSystem::Triclinic:
            M = kIdentity3;
            break;

        case gemmi::CrystalSystem::Monoclinic: {
            const Vec3i b = shortest_along(axes.front().dir);
            // The lattice vectors perpendicular to the two-fold are exactly those it negates, so the
            // plane they span is the kernel of (R + I) - exact, and NOT the coordinate vectors
            // orthogonal to the axis, which is a different set on a non-orthogonal basis. (R + I)
            // has rank one there, so any non-zero row of it is the plane's reciprocal normal.
            Mat3i s = axes.front().rot;
            s[0] += 1; s[4] += 1; s[8] += 1;
            Vec3i normal{0, 0, 0};
            for (int i = 0; i < 3; i++)
                if (s[3*i] != 0 || s[3*i+1] != 0 || s[3*i+2] != 0) {
                    normal = {s[3*i], s[3*i+1], s[3*i+2]};
                    break;
                }
            Vec3i p{}, q{};
            if (!PlaneBasis(normal, metric, p, q))
                return std::nullopt;
            if (Det3(FromRows(p, b, q)) < 0)
                for (int t = 0; t < 3; t++) q[t] = -q[t];

            // Any unimodular pair drawn from that plane is a valid a and c. They differ in how
            // oblique beta comes out and in whether the centring then reads C or I - which are the
            // same lattice in two settings, not two lattices. Of those two namings only C is the
            // REFERENCE setting, and the space-group search this lattice is offered to enumerates
            // reference settings only, so a lattice named I carries a centring no candidate of its
            // point group can match and the promotion it earned is refused for its name. So the
            // least oblique cell is taken among the P and C namings; I is accepted only if the plane
            // offers no other, which would otherwise lose the lattice altogether. An oblique cell is
            // a real cost downstream, where the constrained refinement bounds the cell angles at 30
            // and 150 degrees, so the obliquity is still minimised - within the naming, not across it.
            double best_beta = 1e30, best_len = 1e30;
            bool found = false;
            const auto search = [&](bool allow_i) {
              for (int i = -2; i <= 2; i++)
                for (int j = -2; j <= 2; j++)
                  for (int k = -2; k <= 2; k++)
                    for (int l = -2; l <= 2; l++) {
                      if (i * l - j * k != 1)
                          continue;
                      Vec3i a{}, c{};
                      for (int t = 0; t < 3; t++) {
                          a[t] = i * p[t] + j * q[t];
                          c[t] = k * p[t] + l * q[t];
                      }
                      const double len = metric.Dot(a, a) + metric.Dot(c, c);
                      const double cos_beta = metric.Dot(a, c) / (metric.Len(a) * metric.Len(c));
                      const double beta = std::acos(std::clamp(std::fabs(cos_beta), 0.0, 1.0));
                      const double obtuse = 180.0 - beta * 180.0 / PI;   // beta stated obtuse
                      if (found && (obtuse > best_beta + 1e-6 ||
                                    (obtuse > best_beta - 1e-6 && len >= best_len - 1e-6)))
                          continue;
                      const Mat3i cand = FromRows(a, b, c);
                      int d = 0;
                      const auto t = CentringTranslations(cand, d);
                      const char ce = CentringSymbol(t, d);
                      if (ce != 'P' && ce != 'C' && !(allow_i && ce == 'I'))
                          continue;                  // an A- or I-centred naming of the same lattice
                      best_beta = obtuse; best_len = len; M = cand; found = true;
                    }
            };
            search(false);
            if (!found)
                search(true);
            if (!found)
                return std::nullopt;
            // beta obtuse: negating a and b keeps the handedness, keeps b on the two-fold, and leaves
            // every half-integer centring vector where it was.
            {
                const Vec3i a{M[0], M[1], M[2]}, c{M[6], M[7], M[8]};
                if (metric.Dot(a, c) > 0)
                    for (int t = 0; t < 6; t++) M[t] = -M[t];
            }
            break;
        }

        case gemmi::CrystalSystem::Orthorhombic: {
            std::vector<Vec3i> u;
            for (const Axis &ax : axes)
                u.push_back(shortest_along(ax.dir));
            if (u.size() != 3)
                return std::nullopt;
            std::sort(u.begin(), u.end(), [&](const Vec3i &a, const Vec3i &b) {
                return metric.Dot(a, a) < metric.Dot(b, b);
            });
            M = FromRows(u[0], u[1], u[2]);
            break;
        }

        case gemmi::CrystalSystem::Tetragonal: {
            const auto four = principal_of_order(4);
            if (!four)
                return std::nullopt;
            const Vec3i c = shortest_along(four->dir);
            const auto in_plane = perpendicular_two_folds(four->dir);
            if (in_plane.empty())
                return std::nullopt;
            const Vec3i a = in_plane.front();
            const Vec3i b = MulVec(four->rot, a);       // the 4-fold carries a onto b
            M = FromRows(a, b, c);
            break;
        }

        case gemmi::CrystalSystem::Hexagonal:
        case gemmi::CrystalSystem::Trigonal: {
            const auto principal = system == gemmi::CrystalSystem::Hexagonal ? principal_of_order(6)
                                                                             : principal_of_order(3);
            if (!principal)
                return std::nullopt;
            const Vec3i c = shortest_along(principal->dir);
            const auto in_plane = perpendicular_two_folds(principal->dir);
            if (in_plane.empty())
                return std::nullopt;
            const Vec3i a = in_plane.front();
            // gamma must come out 120, so b is a turned by the THREE-fold, not by the six-fold.
            const Mat3i three = principal->order == 6 ? MatMul(principal->rot, principal->rot)
                                                      : principal->rot;
            const Vec3i b = MulVec(three, a);
            M = FromRows(a, b, c);
            break;
        }

        case gemmi::CrystalSystem::Cubic: {
            std::vector<Vec3i> u;
            for (const Axis &ax : axes)
                if (ax.order == 4)
                    u.push_back(shortest_along(ax.dir));
            std::sort(u.begin(), u.end());
            u.erase(std::unique(u.begin(), u.end()), u.end());
            if (u.size() != 3) {
                // 23 has no four-folds; its conventional axes are the three two-folds.
                u.clear();
                for (const Axis &ax : axes)
                    if (ax.order == 2)
                        u.push_back(shortest_along(ax.dir));
                std::sort(u.begin(), u.end());
                u.erase(std::unique(u.begin(), u.end()), u.end());
            }
            if (u.size() != 3)
                return std::nullopt;
            M = FromRows(u[0], u[1], u[2]);
            break;
        }
        default:
            return std::nullopt;
    }

    if (Det3(M) == 0)
        return std::nullopt;
    if (Det3(M) < 0) {                          // keep the basis right-handed
        for (int j = 0; j < 3; j++)
            M[6 + j] = -M[6 + j];
    }

    int den = 0;
    auto trans = CentringTranslations(M, den);
    char centring = CentringSymbol(trans, den);

    // Bring the answer into the conventional setting of its Bravais class: monoclinic and
    // orthorhombic centrings are named C, and a rhombohedral lattice is described obverse. Each is a
    // relabelling of the same lattice by a unimodular matrix, so the cell it names is unchanged.
    auto apply = [&](const Mat3i &u) {
        M = MatMul(u, M);
        trans = CentringTranslations(M, den);
        centring = CentringSymbol(trans, den);
    };
    if (system == gemmi::CrystalSystem::Orthorhombic) {
        // All three axes are equivalent, so name the centred face ab by permuting them, then put the
        // two axes the centring does not pin back in length order.
        if (centring == 'A')
            apply({0, 1, 0, 0, 0, 1, 1, 0, 0});           // a,b,c -> b,c,a
        else if (centring == 'B')
            apply({1, 0, 0, 0, 0, 1, 0, -1, 0});          // a,b,c -> a,c,-b
        const Vec3i a{M[0], M[1], M[2]}, b{M[3], M[4], M[5]};
        if (metric.Dot(a, a) > metric.Dot(b, b))
            apply({0, 1, 0, 1, 0, 0, 0, 0, -1});          // a <-> b
    }
    if (centring == 'r')
        apply({-1, 0, 0, 0, -1, 0, 0, 0, 1});             // reverse -> obverse

    if (centring == '?' || centring == 'r')
        return std::nullopt;

    LePageResult r;
    r.system = system;
    r.centering = centring;
    r.reindex = ToMat33(M);
    r.primitive_reduced = L_red;
    r.conventional = L_red.Multiply(r.reindex);
    r.max_obliquity_deg = worst_obliquity;
    r.n_operators = static_cast<int>(group.size());
    return r;
}