\documentclass[12pt]{amsart}
%\oddsidemargin -0.7in
%\evensidemargin \oddsidemargin
%\textheight 8in
%\textwidth 11in

\newcommand{\ad}{\operatorname{ad}}
\newcommand{\tr}{\operatorname{tr}}

\begin{document}
\large

\noindent
\[ x+y-\log e^y e^x = (1-e^{-\ad x})F + (e^{\ad y}-1)G \]

\noindent
\[
  \tr(\ad x)\partial_x F + \tr(\ad y)\partial_y G
  = \frac12\tr\left(
    \frac{\ad x}{e^{\ad x}-1} + \frac{\ad y}{e^{\ad y}-1}
    - \frac{\ad z}{e^{\ad z}-1} - 1
  \right)
\]

\[
  \text{with }z=\log e^x e^y
\]

\end{document}