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The time-ordered exponential of a time-dependent matrix $𝖠\left(t\right)$ is defined as the function of $𝖠\left(t\right)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $𝖠\left(t\right)$. The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by $*$. Yet, the existence of such inverses, crucial to...