  The maximal algebras of scalar Toeplitz matrices are known to be formed by
generalized circulants. The identification of algebras consisting of block
Toeplitz matrices is a harder problem, that has received little attention up to
now. We consider the case when the block entries of the matrices belong to a
commutative algebra $ \mathcal{A} $. After obtaining some general results, we
classify all the maximal algebras for certain particular cases of $
\mathcal{A}$.
