  We give a simplified exposition of Kummert's approach to proving that every
matrix-valued rational inner function in two variables has a minimal unitary
transfer function realization. A slight modification of the approach extends to
rational functions which are isometric on the two-torus and we use this to give
a largely elementary new proof of the existence of Agler decompositions for
every matrix-valued Schur function in two variables. We use a recent result of
Dritschel to prove two variable matrix-valued rational Schur functions always
have finite-dimensional contractive transfer function realizations. Finally, we
prove that two variable matrix-valued polynomial inner functions have transfer
function realizations built out of special nilpotent linear combinations.
