Freudenthal magic square

Description

The four real normed division algebras are $\R$, $\C$, $\mathbb{H}$ (quaternions) and $\mathbb{O}$ (octonions). Let $A$ and $B$ be two of them, of dimensions $a$ and $b$. Let $J = H_3(B)$ be the Jordan algebra of $3 \times 3$ Hermitian matrices over $B$, of dimension $3 + 3b$; for $B = \mathbb{O}$ it is the Albert algebra. Let $A_0$ and $J_0$ be the subspaces of elements of trace zero, of dimensions $a - 1$ and $3b + 2$. The Tits construction gives a Lie algebra $$L(A, B) = \operatorname{Der}(A) \oplus (A_0 \otimes J_0) \oplus \operatorname{Der}(J),$$ where $\operatorname{Der}$ denotes the algebra of derivations. The magic square lists the complexifications of these Lie algebras, by the type of their Dynkin diagram:

Check of dimensions: $\operatorname{Der}(A)$ has dimension $0, 0, 3, 14$, and $\operatorname{Der}(H_3(B))$ has dimension $3, 8, 21, 52$ for $B = \R, \C, \mathbb{H}, \mathbb{O}$. For $A = B = \mathbb{O}$ this gives $14 + 7 \cdot 26 + 52 = 248$, the dimension of $E_8$. For $A = \mathbb{H}$ and $B = \mathbb{O}$ it gives $3 + 3 \cdot 26 + 52 = 133$, the dimension of $E_7$. For $A = \R$ the middle term vanishes and $L(\R, B) = \operatorname{Der}(H_3(B))$; for $B = \mathbb{O}$ this is $F_4$ of dimension $52$.