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

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 $n\ge 3$.
