Magic squares and matrix models of Lie algebras
C H Barton, A Sudbery
TL;DR
The paper investigates representing exceptional simple Lie algebras as matrix models over composition algebras, with a focus on octonionic realizations via the Tits–Freudenthal magic square. It presents two symmetric reformulations (Vinberg and triality-based) that illuminate the square's structure and extends the construction to $n\times n$ matrices for associative base algebras, including maximal compact subalgebras and Freudenthal geometries. It also develops $n=2$ and CK-type generalizations (Santander–Herranz) that unify a wide class of real forms, linking derivations of Jordan algebras to classical matrix Lie algebras. Overall, the work provides explicit isomorphisms $L_3(\mathbb{K}_1,\mathbb{K}_2)$ and its variants with $\mathfrak{sa}$, $\mathfrak{sl}$ and $\mathfrak{sp}$-type algebras, offering octonionic and split-algebra perspectives on $F_4$, $E_6$, $E_7$, and $E_8$ in a unified algebraic framework.
Abstract
This paper is concerned with the description of exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the Tits-Freudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a construction in terms of a Jordan algebra of hermitian 3x3 matrices (Tits) or various plane and other geometries (Freudenthal). We present alternative constructions of the magic square which explain its symmetry, and show explicitly how the use of split composition algebras leads to analogues of the matrix Lie algebras su(3), sl(3) and sp(6). We adapt the magic square construction to include analogues of su(2), sl(2) and sp(4) for all real division algebras.