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I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact true!).

Is there is a particularly nice proof in the case of $SU(n)$ that would also be great!

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    $\begingroup$ Given a compact connected Lie group $G$ with Lie algebra $\mathfrak g$, the space of bi-invariant one forms is isomorphic to the space of $Ad(G)$-invariant vectors in the dual of $\mathfrak g$; for s semi-simple connected compact group $G$, such vectors do not exist; if $G$ is a torus, then they do. $\endgroup$
    – Venkataramana
    Commented Aug 21, 2014 at 3:11

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Use conjugation to rotate the one-form at the unit. Use simplicity of $SU(n)$ to show that one can rotate the one-form at the unit in any direction. That, combine with bi-invariance, would imply that the one-form is zero.

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