Transposition symmetry of the magic square

Statement

For real composition algebras $A,B$ , exchanging the two derivation factors and applying $a\otimes b\mapsto b\otimes a$ entrywise gives a Lie algebra isomorphism

\mathfrak M(A,B)\cong\mathfrak M(B,A)

in the Vinberg construction. Consequently the compact magic square is symmetric across its diagonal.

Verification

The tensor flip preserves multiplication, conjugation and matrix trace. It exchanges the two terms in the tensor derivation correction. It therefore preserves each of the three bracket rules, and applying it twice gives the identity.

What the symmetry identifies

The ordered inputs remain distinct catalogue data. For example, the constructions from $(\mathbb R,\mathbb O)$ and $(\mathbb O,\mathbb R)$ both yield compact $\mathfrak f_4$ . This identifies their constructed Lie algebras; it does not assert an algebra isomorphism $\mathbb R\cong\mathbb O$ .

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §4.2, equation (4.19) and Theorem 4.3. 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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Transposition symmetry of the magic square

Statement

Construction and checks

Octonions and triality

Real-form convention

Machine-readable construction

References