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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

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

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

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