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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.

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    What do you mean by "cross sections" or "projections" in this context? – 2017-02-27
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    So $GL_2(\mathbf{R})$ is a subset of $\mathbf{R}^4$, and if we consider the projection of this manifold down into $\mathbf{R}^3$, what does it look like? – 2017-02-27

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

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    Thank you for pointing this out. I guess I should be referring to the groups which are not 4 dimensional. – 2017-02-28