Is there an easy way to rewrite 5x5 matrix multiplications in terms of 4x4 (or smaller) matrices?

Background: I'm trying to implement 4D projections (5x5 matrix multiplications in homogenous coordinates) in the GPU, which has built-in support for 4x4 matrices but does not have 5x5 matrices. I could implement 5x5 matrices manually, but it would run slower than if I took advantage of built-in matrix types.

Are there any nice decompositions of 5x5 matrix multiplication into multiplications of smaller matrices?