Triality decomposition of the magic square

Statement

For real division composition algebras $A,B$ , the magic-square Lie algebra admits the decomposition

\mathfrak M(A,B)\cong\operatorname{tri}(A)\oplus \operatorname{tri}(B)\oplus(A\otimes_{\mathbb R}B)^{\oplus3}

as real vector spaces, with the two triality algebras forming commuting Lie subalgebras. Their three components act on the three tensor copies; the brackets among these copies couple the summands.

Reading the dimensions

Writing $t(A)=\dim\operatorname{tri}(A)$ , this gives

\dim\mathfrak M(A,B)=t(A)+t(B)+3\dim(A)\dim(B).

The values $t=0,2,9,28$ produce all sixteen dimensions of the compact square. In particular the octonion–octonion entry has dimension $28+28+3\cdot64=248$ .

Lie sums versus vector sums

The tensor summands are not commuting Lie ideals. Replacing the construction bracket with a componentwise direct-sum bracket would destroy the exceptional algebras. Compare the genuine direct sum at the complex–complex entry.

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §4.3, Theorem 4.4 and equations (4.22)–(4.26), pp. 21–22. Paper.