Tits decomposition of the magic square

Statement

For $A,B\in\{\mathbb R,\mathbb C,\mathbb H,\mathbb O\}$ , the magic-square Lie algebra has a real-vector-space decomposition

\mathfrak M(A,B)\cong\operatorname{Der}(A)\oplus \operatorname{Der}(H_3(B))\oplus \bigl(\operatorname{Im}A\otimes_{\mathbb R}H_3(B)_0\bigr).

Here $H_3(B)$ has Jordan product $X\circ Y=(XY+YX)/2$ , and the subscript $0$ means trace zero. The first two summands form a commuting Lie subalgebra and act on the tensor summand by their natural actions. The whole display is a vector-space decomposition; brackets between tensor elements also have components in the derivation summands.

The real row

Since $\operatorname{Der}(\mathbb R)=0=\operatorname{Im}\mathbb R$ ,

\mathfrak M(\mathbb R,B)\cong\operatorname{Der}(H_3(B))

as Lie algebras. Thus the first row records derivations of cubic Hermitian Jordan algebras.

Dimension check

If $a=\dim_{\mathbb R}A$ , $b=\dim_{\mathbb R}B$ , then

\dim\mathfrak M(A,B)=\dim\operatorname{Der}(A) +\dim\operatorname{Der}(H_3(B))+(a-1)(2+3b).

This checks dimensions but does not determine the Lie bracket or its real form.

References

John C. Baez, “The Octonions,” §4.3, the Tits decomposition following Table 5. Checked section.