It is well known that if $G$ is a finite group, then the probability that two elements commutte is either $1$ (if $G$ is abelian) or less than or equal to $\frac58$.

If instead $K$ is a compact group, there exists a unique probability over $K$ that is left-invariant (Haar measure). The same problem as above still makes sense: What is the probability that two elements commutte ? In general, this probability can be non-trivial, because $K$ may have several connected components, being a (semi-)direct product of its neutral component $K_0$ with a finite group. If $K_0$ is abelian (a torus), then we are led back to the finite-case result.

So let me assume that $K=K_0$, that is $K$ is connected. Is it possible that the probability that two elements commutte be non-trivial, namely $0<p<1$ ?