Cross sections/projections of matrix groups

Question

If such images exist, I'm looking for a resource which contains images of the cross sections and or projections of matrix groups in 2 and 3 dimensional space (resp.) (i.e. $SL_2(\mathbf{R})$, $O_2(\mathbf{R}))$ and other Lie subgroups if possible.

Any links would be greatly appreciated.

Answer 1

The projection $GL_2(R) \mapsto R^3$ is surjective: for any $(a,b,c) \in R^3$ there exists $d$ such that $$\det\pmatrix{a&b\\c&d}\ne 0$$