$T$-algebras are non-associative algebras defined by Vinberg in the early 1960's for the purpose of studying homogeneous cones. Vinberg defined a cone $K(\mathcal A)$ for each $T$-algebra $\mathcal A$ and proved that every homogeneous cone is isomorphic to one such $K(\mathcal A)$. We relate each $T$-algebra $\mathcal A$ with a space of linear operators in such a way that $K(\mathcal A)$ is isomorphic to the cone of positive definite self-adjoint operators. Together with Vinberg's result, we conclude that every homogeneous cone is isomorphic to a ``slice'' of a cone of positive definite matrices.
Citation
SIAM J. Optim., 2003, 14:500-506