Past research on matrix subalgebras generated by a given set of generators has one major restriction: most of the literature on the generation of algebras automatically presumes unitality of the generator set, meaning that most proofs are based on the implicit assumption of a unit element among the given matrices; this (often hidden) prerequisite rules out an application of all respective theorems to cases where an arbitrary generator set (without the guarantee of an identity matrix) is provided.
A specific example is the recent paper Markova. O. V. & Novochadov, D. Yu., "Generating Systems of the Full Matrix Algebra That Contain Nonderogatory Matrices", J. Math. Sci. 262, 99–107 (2022) with online link https://doi.org/10.1007/s10958-022-05802-2 that gives a necessary and sufficient condition (strongly connected digraph of the generator set interpreted as adjacency matrices) for the generation of the full matrix algebra. Could this method be modified without the assumption of an identity among the generators $X\in \mathbb{C}^{nxn}$ - or equivalently without allowing the monomial $p_0 X^0$ for generating polynomials $p(X)=\sum_ip_i X^i$ ?
Another related question would be why basically all existing research relies on the inclusion of an identity - to me this standard seems to be a rather restrictive prerequisite that is not fulfilled in many applications. Could someone shed some light on this necessity/convention?