Exotic matrix models: the albert Algebra and the spin factor

The matrix models attached to real symmetric matrices and the complex/quaternionic Hermitian matrices have been studied by many authors. These models correspond to three of the simple formally real Jordan algebras over $\mathbb R$. Such algebras were classified by Jordan, von Neumann, and Wigner in the 30s, and apart from these three there are two others: (i) the spin factor $\mathbb S=\mathbb S_{1,n}$, an algebra built on $\mathbb R^{n+1}$, and (ii) the Albert algebra $\mathbb A$ of $3\times3$ Hermitian matrices over the octonions $\mathbb O$. In this paper we investigate the matrix models attached to these remaining cases.