Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL_3$ to $L\cup L'$. Can you tell in what way does $SL_3$ act on $L \cup L'$? What is the stabilizer of $L \cup L'$?

One way to describe this, that fits into various larger patterns, is as a minimal parabolic intersected with its conjugate by a simple rootreflection, and with that reflection adjoined. In coordinates: take lines $x$axis and $y$axis. The uppertriangular matrices $P$ form a standard minimal parabolic. The positive simple roots are $diag(a_1,a_2,a_3)\rightarrow a_1/a_2$ and $a_2/a_3$. The corresponding reflections are $$ \sigma_1=\pmatrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 } \hskip30pt \sigma_2=\pmatrix{ 1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0 } $$ The stabilizer of the union of the two lines is the group generated by $P\cap \sigma_1P\sigma_1$ and $\sigma_1$, the latter interchanging the two lines, the former subgroup stabilizing both lines individually. Perhaps this is not entirely satisfying, but it is a structural description. 

