Extension of commutator-calculus identities to non-symmetric matrices

Let $d$ be a positive integer, let $\mathbf{G}\in\mathbb{C}^{d\times d}$, and let $\mathbf{A}\in\mathbb{C}^{d\times d}$ be such that $\log\mathbf{A}$ exists. Suppose Definition~ is suitably generalized so that the symbols and operators occurring in the stated identities are defined for these matrices. Extension conjecture. All identities of Theorem~, Lemma~, and Corollaries~ and, except the relations and, continue to hold for every such $\mathbf{G}$ and $\mathbf{A}$. The conjecture proposes that the commutator-calculus formulas do not fundamentally depend on symmetry or spectral decomposition, while the two excluded relations use symmetry to simplify the condition on the commutator. Whether the required generalization of Definition~ exists and preserves all these identities is left to future work.