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 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 L_{1,n}, an algebra built on R^{n+1}, and (ii) the Albert algebra A of 3 by 3 Hermitian matrices over the octonions. In this paper we investigate the matrix models attached to these remaining cases.