Inference for L orthogonal models

A mixed model Y^o = ∑^m _i=1 X_iβ_i + ∑_i=m+1 ^w X_iβ̃ + e is orthogonal when the matrices Mi = X_iX_i ^T, i = 1,...,w, commute. The vectors β^c1 ₁,...,β^cm _m are fixed vectors and the β^cm+1 _m+1,...,β^cw _w and eⁿ are random. For these models we have very interesting results namely we have UMVUE for the relevant parameters when normality is assumed. We now intend to generalize that class of models taking Y = L(∑_i=1 ^mX_iβ_i + ∑_i=m+1 ^w X_iβ̃ + e with L a matrix whose column vectors are linearly independent.