adjoint representations I am trying to work out the adjoint representations of
$$H=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right),
X = \left(
\begin{array}{cc}
0 & 1 \\
0 & 0
\end{array}
\right),
Y = \left(
\begin{array}{cc}
0 & 0 \\
1 & 0
\end{array}
\right) .$$
but I am not exactly sure how to do it.
So far I have worked out 
(ad$H$)($H$) = $0$
(ad$H$)($X$) = $\left(\begin{array}{cc}
0 & 2 \\
0 & 0
\end{array}
\right)$
(ad$H$)($Y$) = $\left(\begin{array}{cc}
0 & 0 \\
-2 & 0
\end{array}
\right)$
 A: Let
$$ E_{11}=\left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right),
E_{12}=\left( \begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right),
E_{21}=\left( \begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right),\quad \text{and}\quad
E_{22}=\left( \begin{array}{cc} 0 & 0\\ 0 & 1 \end{array}\right).$$
This is a basis of $\mathbb M_{2}$.
For each pair $(i,j)$ express $adH(E_{ij})$ as a linear combination of matrices from the basis. For instance
$$ adH(E_{12})=2 E_{12}.$$
More generally,
$$ adH(E_{ij})=\alpha_{1}^{(ij)}E_{11}+\alpha_{2}^{(ij)}E_{12}+\alpha_{3}^{(ij)}E_{21}+\alpha_{4}^{(ij)}E_{22}.  $$
Then $ad H$ has a matrix representation (with respect to the above basis):
$$ adH=\left( \begin{array}{cccc} \alpha_{1}^{(11)} & \alpha_{1}^{(12)} & \alpha_{1}^{(21)} & \alpha_{1}^{(22)}\\ 
\alpha_{2}^{(11)} & \alpha_{2}^{(12)} & \alpha_{2}^{(21)} & \alpha_{2}^{(22)}\\ 
\alpha_{3}^{(11)} & \alpha_{3}^{(12)} & \alpha_{3}^{(21)} & \alpha_{3}^{(22)}\\ 
\alpha_{4}^{(11)} & \alpha_{4}^{(12)} & \alpha_{4}^{(21)} & \alpha_{4}^{(22)}
\end{array} \right). $$ 
