Cartan decomposition of measure of $GL(2, \mathbb{C})$

Question

I am searching for a integration formula

$$ \int\limits_{GL_2(\mathbb{C})} \phi(g) d g = \int\limits_{SU(2)} \int\limits_{SU(2)} \int\limits_{M} \phi(k_1mk_2) \omega(m) d m d k_1 d k_2 ,$$ where $M$ is the subgroup of diagonal matrices. I am in particular interested in an explicit form for the weight $\omega$ preferably with reference.

Answer 1

I found now a reference for $\mathrm{SL}(2,\mathbb{C})$ in Brummelhuis, Koornwinder "The generalized Abel transform" Lemma 5.5.

Answer: $\omega$ is trivial on the center and on a diagonal matrix with entries $e^t$ and $e^{-t}$ a scalar multiple of $|sinh(2t)|^2$. The scalar depends of course on the chosen Haar measure.