Existence of diagonal matrix that transforms invertible matrix in unitary This questions arises from the answer to this previous question, which leaded me to 'relax' the statement that I wanted to prove.
Suppose that a matrix $M\in\mathrm{SL}_2(\mathbb{C})\setminus\{\pm I\}$ has diagonal elements complex conjugates of each other and that their product is less than $1$. That is,
$$M=\begin{pmatrix} a & b\\ c & \bar{a}\end{pmatrix},$$
with $|a|^2<1$ and $\det M=1$.
Also, suppose that there exists a matrix $T\in\mathrm{SL}_2(\mathbb{C})$ such that $T^{-1}MT\in\mathrm{SU}_2$.
Then, I think it is possible to find $T$ diagonal, i.e.,

When can find such $T$ with $T=\mathrm{diag}(p, 1/p)$.

I would like to prove this statement and have tried by contradiction with no success.
A couple of preliminary ideas:


*

*Since there exists a matrix $T\in\mathrm{SL}_2(\mathbb{C})$ such that $T^{-1}MT\in\mathrm{SU}_2$, then $\mathrm{tr}M\in (-2,2)$. This is easy to prove using the properties of the trace and those of unitary matrices.

*Therefore, we can deduce that $\mathrm{Re}(a)\in (-1,1)$. Also, since $|a|^2<1$ and the determinants are equal to $1$, we obtain that $bc<0$.
Since I want to prove that such diagonal matrix can be found, I can argue by contradiction and suppose that $T$ cannot be diagonal, that is,
$$T=\begin{pmatrix} t_{11} & t_{12}\\ t_{21} & t_{22}\end{pmatrix}\in\mathrm{SL}_2(\mathbb{C}),$$
with at most one $t_{ij}=0$, for $i,j\in\{1,2\}$.
However, this doesn't seem to be the right approach. I don't really see how to use those conditions stated above for $M$, in order to deduce something about $T$: the only idea that I've had is to actually compute $T^{-1}MT$, which I know is unitary, but I don't know how to use the other conditions (and their implications).
Ideas or clues are very welcome.

What you want is true, even without the hypothesis that there exists $T$ with $T^{-1}MT\in \mathrm{SU}_2$ (which of course ends up being true afterwards). We can also require $p$ real. 
So we assume that $|a|<1$, $\det M=1$, and nothing else. 
For any $p\ne0$, 
$$
\begin{bmatrix} p&0\\0&1/p\end{bmatrix}\begin{bmatrix} a&b\\ c&\bar a\end{bmatrix}\begin{bmatrix} 1/p&0\\0&p\end{bmatrix}
=\begin{bmatrix} a&b\,p^2\\ c/ p^2&\bar a\end{bmatrix}.
$$
We want this to be  a unitary. We have 
$$
\begin{bmatrix} a&b\,p^2\\ c/  p^2&\bar a\end{bmatrix}^*\begin{bmatrix} a&b\,p^2\\ c/ p^2&\bar a\end{bmatrix}
=\begin{bmatrix}
|a|^2+|c|^2/|p|^4 &\bar a(bp^2+\bar c/\bar p^2)\\
a(\bar b \bar p^2+c/p^2)& |a|^2+|b|^2|p|^4.
\end{bmatrix}
$$
For this latter matrix to be the identity, we need $bp^2+\bar c/\bar p^2=0$. Multiplying by $\bar p^2$, 
$$
0=b|p|^4+\bar c,\ \ \ \text{ so } |p|^4=-\bar c/b=-\frac{(|a|^2-1)/\bar b}b=\frac{1-|a|^2}{|b|^2}.
$$
Also, 
$$
|a|^2+|c|^2/|p|^4=|a|^2+|c|^2/(-\bar c/b)=|a|^2-bc=1
$$
and
$$
|a|^2+|b|^2|p|^4=|a|^2+|b|^2(-\bar c/b)=|a|^2-\overline{bc}=\overline{|a|^2-bc}=1.
$$
So any $p\in\mathbb C$ with 
$$
|p|=\left(\frac{1-|a|^2}{|b|^2}\right)^{1/4}
$$
will work. 
As an example, 
$$
\begin{bmatrix} 1/12^{1/4}&0\\0&12^{1/4}\end{bmatrix}
\begin{bmatrix} 1/2&3\\-1/4&1/2\end{bmatrix}
\begin{bmatrix} 12^{1/4}&0\\0&1/12^{1/4}\end{bmatrix}
=\begin{bmatrix} 1/2&\sqrt3/2\\ -\sqrt3/2&1/2\end{bmatrix}. 
$$
