Linear and quadratic sufficiency and commutativity

Given a mixed model let T be the orthogonal projection matrix on the range space spanned by the mean vector. If the model has variance-covariance matrix σ²V we use commutative Jordan algebras to show that Ty is both linear sufficient and linear complete and that Ty, y'V⁺y with V⁺ the Moore-Penrose inverse of V is quadratic sufficient whenever T and V commute.