Derivation algebras of cubic Hermitian Jordan algebras

Statement

For $A=\mathbb R,\mathbb C,\mathbb H,\mathbb O$ , give $H_3(A)$ the real Jordan product $X\circ Y=(XY+YX)/2$ . Its derivation Lie algebra is respectively

\begin{array}{c|cccc} A&\mathbb R&\mathbb C&\mathbb H&\mathbb O\\\hline \operatorname{Der}_{\mathbb R}(H_3(A))& \mathfrak{so}(3)&\mathfrak{su}(3)&\mathfrak{sp}(3)&\mathfrak f_{4,\mathrm{compact}} \end{array}

The dimensions are $3,8,21,52$ . These are compact real Lie algebras; $\mathfrak{sp}(3)$ here uses quaternionic matrix size three.

Why this is the first row

The Tits decomposition gives $\mathfrak M(\mathbb R,A)=\operatorname{Der}(H_3(A))$ . Thus Jordan products determine the first row of the compact square.

Derivations and automorphisms

Derivations are infinitesimal automorphisms, with commutator bracket. They are not arbitrary Jordan endomorphisms: a derivation obeys a Leibniz rule, while a Jordan homomorphism preserves the product itself.

References

C. H. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” §2, equations (2.19)–(2.23); §3 compact square; §5 Theorem 5.1(a). Paper.
John C. Baez, “The Octonions,” §4.2, Theorem 5. Checked section.

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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Derivation algebras of cubic Hermitian Jordan algebras

Statement

Construction and checks

Octonions and triality

Real-form convention

Machine-readable construction

References