Characterization of Wishart–Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart–Laplace distribution (the distribution of the difference of two independent central Wishart Wp(mi,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Qk(Y)} is independent Laplace–Wishart if and only if in addition to the aforementioned conditions, the images ρk(Σ+) of the Moore–Penrose inverse Σ+ of Σ are mutually orthogonal: ρk(Σ+)ρℓ(Σ+)=0 for k≠ℓ.