# OctaveAlgebra() from GAP gives a strange basis not containing 1
# K: base field
# mu1, mu2, mu3: parameters
OctonionAlgebra := function (K, mu1, mu2, mu3)
    local T;
    T := EmptySCTable (8, Zero(K));

    # multiplication table as given in "octonion" paper, e8 is the unit

    SetEntrySCTable (T, 1, 1, [ mu1, 8]);          # e1*e1 = mu1 e8
    SetEntrySCTable (T, 1, 2, [-1,   3]);          # e1*e2 = -e3
    SetEntrySCTable (T, 1, 3, [-mu1, 2]);          # e1*e3 = -mu1 e2
    SetEntrySCTable (T, 1, 4, [-1,   5]);          # e1*e4 = -e5
    SetEntrySCTable (T, 1, 5, [-mu1, 4]);          # e1*e5 = -mu1 e4
    SetEntrySCTable (T, 1, 6, [ 1,   7]);          # e1*e6 = e7
    SetEntrySCTable (T, 1, 7, [ mu1, 6]);          # e1*e7 = mu1 e6
    SetEntrySCTable (T, 1, 8, [ 1,   1]);          # e1*e8 = e1

    SetEntrySCTable (T, 2, 1, [ 1,   3]);          # e2*e1 = e3
    SetEntrySCTable (T, 2, 2, [ mu2, 8]);          # e2*e2 = mu2 e8
    SetEntrySCTable (T, 2, 3, [ mu2, 1]);          # e2*e3 = mu2 e1
    SetEntrySCTable (T, 2, 4, [-1,   6]);          # e2*e4 = -e6
    SetEntrySCTable (T, 2, 5, [-1,   7]);          # e2*e5 = -e7
    SetEntrySCTable (T, 2, 6, [-mu2, 4]);          # e2*e6 = -mu2 e4
    SetEntrySCTable (T, 2, 7, [-mu2, 5]);          # e2*e7 = -mu2 e5
    SetEntrySCTable (T, 2, 8, [ 1,   2]);          # e2*e8 = e2

    SetEntrySCTable (T, 3, 1, [ mu1,     2]);      # e3*e1 = mu1 e2
    SetEntrySCTable (T, 3, 2, [-mu2,     1]);      # e3*e2 = -mu2 e1
    SetEntrySCTable (T, 3, 3, [-mu1*mu2, 8]);      # e3*e3 = -mu1 mu2 e8
    SetEntrySCTable (T, 3, 4, [-1,       7]);      # e3*e4 = -e7
    SetEntrySCTable (T, 3, 5, [-mu1,     6]);      # e3*e5 = -mu1 e6
    SetEntrySCTable (T, 3, 6, [ mu2,     5]);      # e3*e6 = mu2 e5
    SetEntrySCTable (T, 3, 7, [ mu1*mu2, 4]);      # e3*e7 = mu1 mu2 e4
    SetEntrySCTable (T, 3, 8, [ 1,       3]);      # e3*e8 = e3

    SetEntrySCTable (T, 4, 1, [1,   5]);           # e4*e1 = e5
    SetEntrySCTable (T, 4, 2, [1,   6]);           # e4*e2 = e6
    SetEntrySCTable (T, 4, 3, [1,   7]);           # e4*e3 = e7
    SetEntrySCTable (T, 4, 4, [mu3, 8]);           # e4*e4 = mu3 e8
    SetEntrySCTable (T, 4, 5, [mu3, 1]);           # e4*e5 = mu3 e1
    SetEntrySCTable (T, 4, 6, [mu3, 2]);           # e4*e6 = mu3 e2
    SetEntrySCTable (T, 4, 7, [mu3, 3]);           # e4*e7 = mu3 e3
    SetEntrySCTable (T, 4, 8, [1,   4]);           # e4*e8 = e4

    SetEntrySCTable (T, 5, 1, [mu1,      4]);      # e5*e1 = mu1 e4
    SetEntrySCTable (T, 5, 2, [1,        7]);      # e5*e2 = e7
    SetEntrySCTable (T, 5, 3, [mu1,      6]);      # e5*e3 = mu1 e6
    SetEntrySCTable (T, 5, 4, [-mu3,     1]);      # e5*e4 = -mu3 e1
    SetEntrySCTable (T, 5, 5, [-mu1*mu3, 8]);      # e5*e5 = -mu1 mu3 e8
    SetEntrySCTable (T, 5, 6, [-mu3,     3]);      # e5*e6 = -mu3 e3
    SetEntrySCTable (T, 5, 7, [-mu1*mu3, 2]);      # e5*e7 = -mu1 mu3 e2
    SetEntrySCTable (T, 5, 8, [1,        5]);      # e5*e8 = e5

    SetEntrySCTable (T, 6, 1, [-1,       7]);      # e6*e1 = -e7
    SetEntrySCTable (T, 6, 2, [ mu2,     4]);      # e6*e2 = mu2 e4
    SetEntrySCTable (T, 6, 3, [-mu2,     5]);      # e6*e3 = -mu2 e5
    SetEntrySCTable (T, 6, 4, [-mu3,     2]);      # e6*e4 = -mu3 e2
    SetEntrySCTable (T, 6, 5, [ mu3,     3]);      # e6*e5 = mu3 e3
    SetEntrySCTable (T, 6, 6, [-mu2*mu3, 8]);      # e6*e6 = -mu2 mu3 e8
    SetEntrySCTable (T, 6, 7, [ mu2*mu3, 1]);      # e6*e7 = mu2 mu3 e1
    SetEntrySCTable (T, 6, 8, [ 1,       6]);      # e6*e8 = e6

    SetEntrySCTable (T, 7, 1, [-mu1,         6]);  # e7*e1 = -mu1 e6
    SetEntrySCTable (T, 7, 2, [ mu2,         5]);  # e7*e2 = mu2 e5
    SetEntrySCTable (T, 7, 3, [-mu1*mu2,     4]);  # e7*e3 = -mu1 mu2 e4
    SetEntrySCTable (T, 7, 4, [-mu3,         3]);  # e7*e4 = -mu3 e3
    SetEntrySCTable (T, 7, 5, [ mu1*mu3,     2]);  # e7*e5 = mu1 mu3 e2
    SetEntrySCTable (T, 7, 6, [-mu2*mu3,     1]);  # e7*e6 = -mu2 mu3 e1
    SetEntrySCTable (T, 7, 7, [ mu1*mu2*mu3, 8]);  # e7*e7 = mu1 mu2 mu3 e8
    SetEntrySCTable (T, 7, 8, [ 1,           7]);  # e7*e8 = e7

    SetEntrySCTable (T, 8, 1, [1, 1]);             # e8*e1 = e1
    SetEntrySCTable (T, 8, 2, [1, 2]);             # e8*e2 = e2
    SetEntrySCTable (T, 8, 3, [1, 3]);             # e8*e3 = e3
    SetEntrySCTable (T, 8, 4, [1, 4]);             # e8*e4 = e4
    SetEntrySCTable (T, 8, 5, [1, 5]);             # e8*e5 = e5
    SetEntrySCTable (T, 8, 6, [1, 6]);             # e8*e6 = e6
    SetEntrySCTable (T, 8, 7, [1, 7]);             # e8*e7 = e7
    SetEntrySCTable (T, 8, 8, [1, 8]);             # e8*e8 = e8

    return (AsAlgebraWithOne (K, AlgebraByStructureConstants (K, T)));
end; 




# returns the standard involution on the octonion algebra O
OctonionInvolution := function (O)
    local b, images;
    b := CanonicalBasis (O);
    images := List ([1..7], i -> -b[i]);
    Add (images, b[8]);
    return (LeftModuleHomomorphismByImages (O, O, b, images));
end;