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
I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$.
Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, where $u$ is an element of $SU(2)$ and $s$ is a matrix that is a product of two matrices $a$ and $k...