Non-compact real forms of $\mathfrak {sl}_3(\mathbb C)$ 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 $\mathfrak s$?
 A: 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.
