Suppose $g_1,...,g_n\in\mathbb{M}_{d\times d}(\mathbb{C})$ are matrices and we are interested in finding the smallest matrix Lie algebra containing them, that is, the matrix Lie algebra generated by $g_1,...,g_n$. Specifically, we are interested in obtaining a vector space basis for this algebra. We are assuming the Lie operation is commutation.
Typically, one picks a linearly independent subset of $g_1,...,g_n$ (with the same linear span) and starts taking commutations. Commutations that are linearly independent to the previous elements are appended to this list until a basis is formed.
To avoid doing more work than necessary, since the Jacobi identity will cause checking some commutations to be redundant, one follows a strategy in the checked commutations. The strategy I am interested in are so called Philip Hall bases. See here for a quick reference.
My question is about stopping conditions. The trivial stopping condition is when you have enough elements to span $\mathfrak{gl}_d(\mathbb{C})$. The more interesting stopping condition occurs when the Lie algebra being generated is a proper subalgebra of $\mathfrak{gl}_d(\mathbb{C})$. I have heard rumours that if all P. Hall basis elements of a given depth $m\in\mathbb{N}$ are linearly dependent on P. Hall basis elements from smaller depths $m'<m$, then all P. Hall basis elements with depth $m'>m$ will also be linearly dependant on the elements with depth less than $m$.
Is this true?
