<s>[INST] Writing $I + x(A+B)+ M_2x^2+M_3x^3+\cdots = e^{x(A+B)+x^2R_2+x^3R_3+\cdots}$ Suppose I have two complex matrices $A,B$, of the same dimension I am wondering if it possible to represent for $x \in \mathbb{R}$, $e^{x/2A}e^{xB}e^{x/2A} = e^{x(A+B) + x^2R2+x^3R_3+\cdots}$ for some matrices $R_2,R_3,\ldots$. 
I know that $e^{x/2A}e^{xB}e^{x/2A} = (I+x/2A+x^2/8A +\cdots)(I+xB+x^2/2B + \cdots)(I+x/2A+x^2/8A + \cdots)= I + x(A+B)+ M_2x^2+M_3x^3+\cdots $ for some matrices $\{M_i\}_{i=2}^{\infty}$.
Can I then write
$I + x(A+B)+ M_2x^2+M_3x^3+\cdots = e^{x(A+B)+x^2R_2+x^3R_3+\cdots}$ for some matrices $\{R_i\}_{i=2}^{\infty}$ ?
 [/INST] Just expand: 
$$\log(I + A) = A  - \frac{1}{2} A^2  + \frac{1}{3} A^3  + \ldots $$
(converging if $\|A\| < 1$)
so (say for $|z|<r$ where $\log(e^{Az/2} e^{Bz} e^{Az/2})$ is analytic)
$$ \eqalign{\log(I + M_1 z + M_2 z^2 + M_3 z^3 &+ \ldots) = (M_1 z + M_2 z^2 + M_3 z^3 + \ldots) \cr &- \frac{1}{2} (M_1 z + M_2 z^2 + \ldots)^2 + \frac{1}{3} (M_1 z + \ldots)^3 + \ldots\cr
&= M_1 z + \left(M_2 - \frac{M_1^2}{2}\right) z^2 + \left(M_3 + \frac{M_1M_2 + M_2 M_1 }{2}  + \frac{M_1^3}{3}\right) z^3+\ldots}$$
etc.
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