I want to ask General strategy of the error bound of the matrix exponential. For example, suppose, $A, B$ are finite dimension $n \times n$ matrices with complex coefficients. Using Baker–Campbell–Hausdorff formula or simple calculation. One can show that ($x \geq 0$)

$e^{Ax}e^{Bx}-e^{Bx}e^{Ax}=[A,B]x^2+O(|x^3|).$

But the thing I want is that the clear bounds. I want to find the $c$ such that $||e^{Ax}e^{Bx}-e^{Bx}e^{Ax}-[A,B]x^2|| \leq c|x^3|.$ Here $c$ should be function of $A,B.$

The first method I want to try is to use Taylor's theorem. But the computation is so tedious. Is there any way that I can find the clear expression for $c$?

More generally, given any matrix expression in terms of parameters $x$, one can easily see the first few terms, but is there a more elegant way to control the error bounds? Or in the simplest case, what is the error bounds for Baker–Campbell–Hausdorff formula $e^{Ax}e^{Bx}$ to $n$ order. I mean the clear expression of the constant before $|x|^n$ not just the big O notation.