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Let su(2) be the Lie algebra of SU(2), thought of as a representation of SU(2) by conjugation.

Let S be the rank-2 symmetric tensors over the complex numbers, acted on by SU(2) in the obvious way.

I have seen it asserted that the complexification of su(2) is isomorphic (as a representation) to S, and presumably there is some simple explicit isomorphism that should be obvious. Alas, to me it is not obvious. What is that isomorphism?

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