I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that appears in such expressions. These are collectively referred to as the "Hall basis". [See here ]

I would like to find a shortcut in terms of matrix algebras. In particular I was wondering if there is a matrix algebra of $n$-dimensional matrices in which any member can be written as a sum of commutators of up to degree $d$. I think $n$ will grow exponentially with $d$.

I am trying to use such a method to streamline some calculations in the dynamics of open quantum systems were such expansions (e.g. Magnus expansion ) occur naturally but are hard to work with. I think the solution must have to do with triangular matrices but I might be missing something. Perhaps the same thing can be done with a little overhead by rephrasing in terms of products and nil-potency as opposed to commutators. Any pointers in helpful directions are much appreciated!