139 of themChecks

Every claim this site makes about these algebras is computed before the page is built, and the build refuses to publish if one fails. Here is the list, as of 2026-09-23: 139 checks, 0 failing.

Two files do the work. tools/algebra.py multiplies things out in exact integer arithmetic — the tables, the laws, the Fano lines, the triality form, the magic square. tools/roots.py builds each root system from its definition, finds the simple roots by cutting with a generic hyperplane, multiplies out the Coxeter element, and reads the projection plane off its eigenvector.

Run them yourself:

python3 tools/algebra.py
python3 tools/roots.py

Algebra 99 checks, 0 failing

✓ R: alternative
✓ R: |xy| = |x||y|
✓ R: e0 is the unit
✓ C: alternative
✓ C: |xy| = |x||y|
✓ C: e0 is the unit
✓ H: alternative
✓ H: |xy| = |x||y|
✓ H: e0 is the unit
✓ O: alternative
✓ O: |xy| = |x||y|
✓ O: e0 is the unit
✓ C': alternative
✓ C': |xy| = |x||y|
✓ C': e0 is the unit
✓ H': alternative
✓ H': |xy| = |x||y|
✓ H': e0 is the unit
✓ O': alternative
✓ O': |xy| = |x||y|
✓ O': e0 is the unit
✓ O: composition holds on 800 random pairs
✓ S: composition fails past dimension 8
✓ S: a pair of nonzero sedenions multiplying to zero
✓ O: 7 lines in the table
✓ O: the seven lines cover the seven imaginary units
✓ O: each imaginary unit lies on three lines
✓ C: T(v,psi,chi) is cyclic
✓ H: T(v,psi,chi) is cyclic
✓ O: T(v,psi,chi) is cyclic
✓ C': T(v,psi,chi) is cyclic
✓ H': T(v,psi,chi) is cyclic
✓ O': T(v,psi,chi) is cyclic
✓ O: a reflection flips the sign of T (960 exact cases)
✓ O: and on 1500 random unit octonions, worst miss 2.8e-14
✓ H: t cycles e1 -> e2 -> e3 -> e1
✓ H: t^3 = 1
✓ su(3,R) = tri(R) + 3x1 = 3 = su(2)
✓ su(3,C) = tri(C) + 3x2 = 8 = su(3)
✓ su(3,H) = tri(H) + 3x4 = 21 = sp(3)
✓ su(3,O) = tri(O) + 3x8 = 52 = f4
✓ su(3,C') = tri(C') + 3x2 = 8 = sl(3)
✓ su(3,H') = tri(H') + 3x4 = 21 = sp(6,R)
✓ su(3,O') = tri(O') + 3x8 = 52 = f4(4)
✓ L(R,R) = 0+0+3 = 3 = su(2)
✓ L(R,C) = 0+2+6 = 8 = su(3)
✓ L(R,H) = 0+9+12 = 21 = sp(3)
✓ L(R,O) = 0+28+24 = 52 = f4
✓ L(R,C') = 0+2+6 = 8 = sl(3)
✓ L(R,H') = 0+9+12 = 21 = sp(6,R)
✓ L(R,O') = 0+28+24 = 52 = f4(4)
✓ L(C,R) = 2+0+6 = 8 = su(3)
✓ L(C,C) = 2+2+12 = 16 = 2 su(3)
✓ L(C,H) = 2+9+24 = 35 = su(6)
✓ L(C,O) = 2+28+48 = 78 = e6
✓ L(C,C') = 2+2+12 = 16 = sl(3,C)
✓ L(C,H') = 2+9+24 = 35 = su(3,3)
✓ L(C,O') = 2+28+48 = 78 = e6(2)
✓ L(H,R) = 9+0+12 = 21 = sp(3)
✓ L(H,C) = 9+2+24 = 35 = su(6)
✓ L(H,H) = 9+9+48 = 66 = so(12)
✓ L(H,O) = 9+28+96 = 133 = e7
✓ L(H,C') = 9+2+24 = 35 = sl(3,H)
✓ L(H,H') = 9+9+48 = 66 = sp(6,H)
✓ L(H,O') = 9+28+96 = 133 = e7(-5)
✓ L(O,R) = 28+0+24 = 52 = f4
✓ L(O,C) = 28+2+48 = 78 = e6
✓ L(O,H) = 28+9+96 = 133 = e7
✓ L(O,O) = 28+28+192 = 248 = e8
✓ L(O,C') = 28+2+48 = 78 = e6(-26)
✓ L(O,H') = 28+9+96 = 133 = e7(-25)
✓ L(O,O') = 28+28+192 = 248 = e8(-24)
✓ L(C',R) = 2+0+6 = 8 = sl(3)
✓ L(C',C) = 2+2+12 = 16 = sl(3,C)
✓ L(C',H) = 2+9+24 = 35 = sl(3,H)
✓ L(C',O) = 2+28+48 = 78 = e6(-26)
✓ L(C',C') = 2+2+12 = 16 = 2 sl(3)
✓ L(C',H') = 2+9+24 = 35 = sl(6,R)
✓ L(C',O') = 2+28+48 = 78 = e6(6)
✓ L(H',R) = 9+0+12 = 21 = sp(6,R)
✓ L(H',C) = 9+2+24 = 35 = su(3,3)
✓ L(H',H) = 9+9+48 = 66 = sp(6,H)
✓ L(H',O) = 9+28+96 = 133 = e7(-25)
✓ L(H',C') = 9+2+24 = 35 = sl(6,R)
✓ L(H',H') = 9+9+48 = 66 = so(6,6)
✓ L(H',O') = 9+28+96 = 133 = e7(7)
✓ L(O',R) = 28+0+24 = 52 = f4(4)
✓ L(O',C) = 28+2+48 = 78 = e6(2)
✓ L(O',H) = 28+9+96 = 133 = e7(-5)
✓ L(O',O) = 28+28+192 = 248 = e8(-24)
✓ L(O',C') = 28+2+48 = 78 = e6(6)
✓ L(O',H') = 28+9+96 = 133 = e7(7)
✓ L(O',O') = 28+28+192 = 248 = e8(8)
✓ su(3): 2 + 2 x 3 = 8
✓ sp(3): 7 + 2 x 7 = 21
✓ f4: 22 + 2 x 15 = 52
✓ e6: 30 + 2 x 24 = 78
✓ e7: 49 + 2 x 42 = 133
✓ e8: 92 + 2 x 78 = 248

Root systems 40 checks, 0 failing

✓ G2: 12 roots
✓ G2: Coxeter number 6
✓ G2: rings of 6, 6 on 2 circles
✓ G2: 2 simple roots, rank 2
✓ G2: dim = 12 roots + rank 2 = 14
✓ D4: 24 roots
✓ D4: Coxeter number 6
✓ D4: rings of 18, 6 on 2 circles
✓ D4: 4 simple roots, rank 4
✓ D4: dim = 24 roots + rank 4 = 28
✓ F4: 48 roots
✓ F4: Coxeter number 12
✓ F4: rings of 12, 12, 12, 12 on 4 circles
✓ F4: 4 simple roots, rank 4
✓ F4: dim = 48 roots + rank 4 = 52
✓ E6: 72 roots
✓ E6: Coxeter number 12
✓ E6: rings of 24, 12, 24, 12 on 4 circles
✓ E6: 6 simple roots, rank 6
✓ E6: dim = 72 roots + rank 6 = 78
✓ E7: 126 roots
✓ E7: Coxeter number 18
✓ E7: rings of 18, 18, 18, 18, 18, 18, 18 on 7 circles
✓ E7: 7 simple roots, rank 7
✓ E7: dim = 126 roots + rank 7 = 133
✓ E8: 240 roots
✓ E8: Coxeter number 30
✓ E8: rings of 30, 30, 30, 30, 30, 30, 30, 30 on 8 circles
✓ E8: 8 simple roots, rank 8
✓ E8: dim = 240 roots + rank 8 = 248
✓ e8: a triality split of 84 + 78 + 78 roots
✓ e8: the fixed 84 roots have rank 7 — so(14), and 84 + 8 = 92 with the Cartan
✓ e8: the split respects brackets on all 13440 root sums
✓ e8: 112 + 8 = 120 = so(16), and 128 = one spinor
✓ D4 triality: orthogonal
✓ D4 triality: order3
✓ D4 triality: permutes_roots
✓ D4 triality: 8v_to_8c
✓ D4 triality: 8c_to_8s
✓ D4 triality: 8s_to_8v

What is not checked here

The names in the magic square — which real form each box is — are taken from the paper's Table 6 and from the literature it cites C. H. Barton 2003 J. Evans 2009; what this repository verifies is that the dimension formula reproduces the dimension of the algebra named. The particle assignments on the physics page are not checkable arithmetic at all; they are proposals, attributed where they are made.