If $A \in \mathbb R^{n \times n}$ then I know that $\operatorname{span}\left(\left\{ e^{tA} v \mid t \in \mathbb R \right\}\right) = \operatorname{span}\left(\left\{ v, A v, A^2 v, \dots \right\}\right) = \operatorname{span}\left(\left\{ v, A v, A^2 v, \dots , A^{n-1} v \right\}\right)$.
I'm interested in a generalization of this. How can I explicitly compute in terms of $A, B \in \mathbb R^{n \times n}$ - not necessarily commuting - and $v \in \mathbb R^n$ what is $R(A, B, v) := \operatorname{span}\left(\left\{ e^{tA} e^{tB} v \mid t \in \mathbb R \right\}\right)$?
It's clear that
$$ R(A, B, v) \subseteq \operatorname{span}\left(\left\{ e^{tA} e^{sB} v \mid s, t \in \mathbb R \right\}\right) = \\ \operatorname{span}\left(\left\{ v, B v, \dots, B^{n-1} v, A v, A B v, \dots, A B^{n-1} v, A^2 v, \dots \dots , A^{n-1} B^{n-1} v \right\}\right) $$
by repeated application of the case mentioned at the start of the question, but that is an overapproximation without a clear way of nailing down which subspace is part of the span I'm after.