  We study algebraic properties of full rank 1 algebras in a general framework
and derive a method to verify if one such matrix polynomial sub-algebra is
bispectral. We give two examples illustrating the method. In the first one, we
consider the eigenvalue to be scalar-valued, whereas, in the second one, we
assume it to be matrix-valued. In the former example, we put forth a Pierce
decomposition of that algebra.
