I was experimenting on $e^M$ and found this:
When $ M = \left[ \begin{matrix} 0 & x \\ y & 0 \end{matrix} \right] $
$e^M = \left[ \begin{matrix} cosh(\sqrt{xy}) & \sqrt{x\over{y}}sinh(\sqrt{xy}) \\ \sqrt{y\over{x}}sinh(\sqrt{xy}) & cosh(\sqrt{xy}) \end{matrix} \right]$
Expressing $M = A + B$, where $ A = \left[ \begin{matrix} a & x \\ 0 & a \end{matrix} \right] $ and $ B = \left[ \begin{matrix} -a & 0 \\ y & -a \end{matrix} \right] $, gives
$e^M = e^{A+B} = e^Ae^B = \left[ \begin{matrix} e^a & xe^a \\ 0 & e^a \end{matrix} \right] \left[ \begin{matrix} e^{-a} & 0 \\ ye^{-a} & e^{-a} \end{matrix} \right] = \left[ \begin{matrix} xy+1 & x \\ y & 1 \end{matrix} \right]$
Is this expected? Or am I doing something wrong? Which one is correct?
You are assuming that $$\tag1e^{A+B}=e^Ae^B.$$ This is not true in general; you need $AB=BA$ for $(1)$ to hold.