Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis For a lie algebra $\mathbb{g}  $ we can define the adjoint representation as:
 $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $
I am wondering how to we compute adjoint matrices with respect to a given basis? 

For example, let the lie algebra be $ \mathbb{g}=sl_2(\mathbb{C}) $ with the basis as follows:
$X=\begin{bmatrix} 0&1\\0& 0 \end{bmatrix}$,$Y=\begin{bmatrix} 0&0\\1& 0 \end{bmatrix}$ and $H=\begin{bmatrix} 1&0\\0& -1 \end{bmatrix}$
Let $ ad: \mathbb{g} \rightarrow End(\mathbb{g})  $ be the adjoint representation
How would I compute the matrices $ ad_X $, $ ad_Y $ and $ ad_Z$ relative to the above matrices?
 A: Since $sl_2(\Bbb C)$ is $3$-dimensional as a vector space, you can imagine the basis $\{X,Y,H\}$ of $\mathbb{g}=sl_2(\mathbb{C})$, as 
$\Bigg\{
\left(
\begin{array}{l} 
1 \\ 0 \\ 0 
\end{array}
\right)$,
$\left(\begin{array}{l}
0 \\ 1 \\ 0 
\end{array}\right)$,
$\left(\begin{array}{c} 
0  \\ 0  \\ 1 
\end{array}\right)
\Bigg\}$, i.e. a basis for $\mathbb{R}^3$
For the $X$ generator:
$$ad_X(X)=[X,X]=0, \ \ 
ad_X(Y)=[X,Y]=H, \ \ 
ad_X(H)=[X,H]=-2X$$ 
thus: $$X\mapsto ad_X=\Big[ad_X(X),ad_X(Y),ad_X(H)\Big]=\begin{bmatrix}0 & 0 & -2 \\ 0 & 0 & \ \ \ 0 \\ 0 & 1 & \ \ \ 0 \end{bmatrix}\in End(\mathbb{g})$$ 
Similarly, for the $Y$ generator:
$$
ad_Y(X)=[Y,X]=-H, \ \ 
ad_Y(Y)=[Y,Y]=0, \ \ 
ad_Y(H)=[Y,H]=2Y
$$ 
thus: $$Y\mapsto ad_Y=\Big[ad_Y(X),ad_Y(Y),ad_Y(H)\Big]=\begin{bmatrix} \ \ \ 0 & 0 & 0 \\ \ \ \ 0 & 0 & 2 \\ -1 & 0 & 0 \end{bmatrix}\in End(\mathbb{g})$$ 
And finally, for the $H$ generator:
$$
ad_H(X)=[H,X]=2X, \ \ 
ad_H(Y)=[H,Y]= -2Y, \  \ 
ad_H(H)=[H,H]=0
$$
thus: $$H\mapsto ad_H=\Big[ad_H(X),ad_H(Y),ad_H(H)\Big]=\begin{bmatrix}2 & \ \ \ 0 & 0 \\ 0 & -2 & 0 \\ 0 & \ \ \ 0 & 0 \end{bmatrix}\in End(\mathbb{g})$$ 
