Alternative Cartan Subalgebra of ${\frak sl}{n}$

Question

So i know that for ${\frak sl}{n}$ the diagonal matrices are a Cartan Subalgebra, and i know that all CSA's are conjugate. My question is, what is an explicity example of a conjugate CSA for ${\frak sl}{n}$?

Answer 1

For $n=2$, and the standard basis of $\mathfrak{sl}_2$, we can write down easily two different Cartan subalgebras, namely $$ \mathfrak{h}_1=\biggl\{\begin{pmatrix} a & 0 \cr 0 & -a \end{pmatrix}\biggr\};\quad \mathfrak{h}_2=\biggl\{\begin{pmatrix} 0 & b \cr -b & 0 \end{pmatrix}\biggr\} $$ They are not conjugated by an inner automorphism (over $\mathbb{R}$). This can be easily generalized to all $n\ge 3$.

Answer 2

Take any automorphism $\varphi$ of $\mathfrak{sl}_n$ and consider $\varphi(\mathfrak{h})$, where $\mathfrak h$ is the standard Cartan subalgebra (that is, the diagonal matrices).

For instance, in the case of $\mathfrak{sl}_2(\mathbb{C})$, you can take$$\varphi\begin{pmatrix}a&b\\c&-a\end{pmatrix}=\begin{pmatrix}a+c&-2a+b-c\\c&-a-c\end{pmatrix},$$in which case$$\varphi(\mathfrak{h})=\left\{\begin{pmatrix}a&-2a\\0&-a\end{pmatrix}\,\middle|\,a\in\mathbb{C}\right\}.$$