Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Let \(G\) be the Lie subgroup of \(\GL_3(\R)\) that consists of the following
matrices:
\[
  \begin{pmatrix}
    a & b & x\\
    c & d & y\\
    0 & 0 & 1
  \end{pmatrix}
\]
where \(ad-bc=1\). Find a basis for the Lie algebra of \(G\), considered as a Lie
subalgebra of \(M_{3\times3}(\R)\).
\item[(b)] Let \(\mathfrak{gl}_2(\R)\) be the Lie algebra of \(\GL_2(\R)\). For
\[
  \begin{pmatrix}2&1\\3&-2\end{pmatrix},
  \begin{pmatrix}1&3\\0&1\end{pmatrix}
  \in\mathfrak{gl}_2(\R)
\]
find
\[
  \left[
    \begin{pmatrix}2&1\\3&-2\end{pmatrix},
    \begin{pmatrix}1&3\\0&1\end{pmatrix}
  \right].
\]
\end{enumerate}