Forty-nine boxes, one ruleThe Square

A Lie algebra is the bookkeeping of a continuous symmetry: the list of independent ways to nudge something, and the rule for what happens when you nudge twice in different orders. Rotations in three dimensions give a three-dimensional one. Most of them come in four tidy families that run forever. Five do not: g2, f4, e6, e7, e8. Those five are the exceptional ones, and they exist because the octonions do.

Freudenthal and Tits found the square in the sixties H. Freudenthal 1964 J. Tits 1966. The construction is: take the triality algebra of 𝔸, the triality algebra of 𝔹, and three copies of 𝔸 ⊗ 𝔹 — one for the vector, one for each spinor.

Every dimension in the table is computed from that formula by tools/algebra.py and then checked against the known dimension of the algebra named in the box. All forty-nine agree.

Look down that column. 52, 78, 133, 248 — the whole exceptional series except g2, produced by one arithmetic rule from the octonions and one other algebra. g2 is the odd one out because it is not in the square at all: it is the symmetry group of the octonions, the fourteen-dimensional set of ways to relabel them that leaves every product where it was Lisi 2026 §11.

And the square is symmetric, which it had no right to be. L(𝔸, 𝔹) and L(𝔹, 𝔸) are built differently and come out the same. That is the part that earns the name.