Vinberg magic-square construction

The Vinberg magic-square Lie algebra of two real unital composition algebras $A,B$ is

\mathfrak M(A,B)=\operatorname{Der}(A)\oplus\operatorname{Der}(B) \oplus\mathfrak{sa}_3(A\otimes_{\mathbb R}B),

as a real vector space. Here $\operatorname{Der}(A)$ consists of linear maps satisfying $D(xy)=D(x)y+xD(y)$ , with commutator bracket, and $\mathfrak{sa}_3$ consists of conjugate-skew matrices of trace zero. Tensor multiplication and conjugation act factorwise. The first two summands commute and act entrywise on the last; for matrices $X,Y$ , the remaining bracket is

[X,Y]=(XY-YX)_0+\frac13\sum_{i,j}D_{X_{ij},Y_{ji}}, \qquad Z_0=Z-\frac{\operatorname{tr}Z}{3}I.

For elements of either factor put

D_{a,c}=[L_a,L_c]+[L_a,R_c]+[R_a,R_c],

where $L_a(x)=ax$ , $R_a(x)=xa$ . Extend to tensor arguments bilinearly by

D_{a\otimes b,c\otimes d} =\langle b,d\rangle D_{a,c}+\langle a,c\rangle D_{b,d}, \qquad \langle x,y\rangle=\operatorname{Re}(x\bar y).

These brackets satisfy the Lie algebra axioms.

Compact inputs

Taking each input from $\mathbb R,\mathbb C,\mathbb H,\mathbb O$ gives the compact Freudenthal magic square. The derivation correction is essential: an uncorrected matrix commutator over octonionic tensor coefficients does not define this construction.

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §4.2, equations (4.19)–(4.21) and Theorem 4.3, pp. 19–21. Paper.

Input	$\mathbb R$	$\mathbb C$	$\mathbb H$	$\mathbb O$
$\mathbb R$	$\mathfrak{so}(3)$	$\mathfrak{su}(3)$	$\mathfrak{sp}(3)$	$\mathfrak f_4$
$\mathbb C$	$\mathfrak{su}(3)$	$\mathfrak{su}(3)\oplus\mathfrak{su}(3)$	$\mathfrak{su}(6)$	$\mathfrak e_6$
$\mathbb H$	$\mathfrak{sp}(3)$	$\mathfrak{su}(6)$	$\mathfrak{so}(12)$	$\mathfrak e_7$
$\mathbb O$	$\mathfrak f_4$	$\mathfrak e_6$	$\mathfrak e_7$	$\mathfrak e_8$

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Vinberg magic-square construction

Construction and checks

Octonions and triality

Real-form convention

Machine-readable construction

References