Non-compact real forms of $\mathfrak {sl}_3(\mathbb C)$

Question

I am looking the non-compact real forms $\mathfrak s$ of $\mathfrak {sl}_3(\mathbb C)$ with $\mathfrak {sl}_3(\mathbb C)=\mathfrak s\oplus i\mathfrak s$? What are the corresponding Lie groups of $\mathfrak s$?

Answer 1

Up to isomorphism, there are three real forms of $\mathfrak{sl}_3(\mathbb C)$:

$\mathfrak{g}_1 = \mathfrak{sl}_3(\mathbb R) = \lbrace \begin{pmatrix} a & c & e\\ f & b & d\\ h & g & -a-b \end{pmatrix} : a, ..., h \in \mathbb{R} \rbrace$;

$\mathfrak{g}_2 = \mathfrak{su}_{1,2} := \lbrace \begin{pmatrix} a+bi & c+di & ei\\ f+gi & -2bi & -c+di\\ hi & -f+gi & -a+bi \end{pmatrix} : a, ..., h \in \mathbb{R} \rbrace$;

$\mathfrak{g}_3 = \mathfrak{su}_{3} := \lbrace \begin{pmatrix} ia & c+di & g+hi\\ -c+di & ib & e+fi\\ -g+hi & -e+fi & -ai-bi \end{pmatrix} : a, ..., h \in \mathbb{R} \rbrace$.

The first one is the split real form, the second one is quasi-split but not split, and the third one is the compact form. I am pretty sure "the" corresponding Lie groups of the non-compact forms would be denoted $SL_3(\mathbb R)$ resp. $SU(1,2)$.

For everything you want to know about these forms and their representation theory, look at this answer (part "A slightly different example"). For real forms of $\mathfrak{sl}_n$ in general, learn to read this table.