# Spanning the Lie Algebra of $SL_{2}(\mathbf{R})$

What is a sufficient criteria for testing whether or not a set of matrices span the Lie algebra of $SL_{2}(\mathbf{R})$?

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Finding three linearly independent ones? – Jyrki Lahtonen Oct 12 '11 at 7:57

One condition is that they not have a common one-dimensional invariant subspace in $\mathbb C^2$.