Adjoint matrix in $\mathbb{so_3}$

Question

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & \end{bmatrix}$ and $X_3=\begin{bmatrix} & 1& \\ =1& & \\ & & 0 \end{bmatrix}$

Write down the matrix $ad_{X_1}$ with respect to the above basis $\{X_1, X_2, X_3 \}$

I know that $[X_1, X_2]=X_3$, $[X_2, X_3]=X_1$ and $[X_3, X_1]=X_2$

Also, $\mathbb{so_3}=\{ X \in \mathbb{sl_3(R)} : X + {^t}X=0 \}$ which is of dimension $3$

How can we use these facts (or others?) to compute, say, $ad_{X_1}$?

I think it will be a $3 \times 3$ matrix

Answer 1

Identifying $X_1,X_2,X_3$ with the standard basis $\Bigg\{ \left( \begin{array}{l} 1 \\ 0 \\ 0 \end{array} \right)$, $\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right)$, $\left(\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right) \Bigg\}$ of $\mathbb{R}^3$, we get: $$X_1\mapsto ad_{X_1}=\Big[ad_{X_1}(X_1),ad_{X_1}(X_2),ad_{X_1}(X_3)\Big]=\Big[[X_1,X_1],[X_1,X_2],[X_1,X_3]\Big]= \\ =\Big[0,X_3,-X_2\Big]= \left( \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array} \right) \in End(so(3)) $$