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 $SL_{2}(\mathbf{R})$?

There should be as many matrices as the algebra, be elements of the algebra and they should be linearly independent.

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