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Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two involutions. If $D$ is commutative, then every noncentral element in $\mathrm{SL}_2(D)$ is similar to $\begin{pmatrix}0&1\\-1&a\end{pmatrix}$, which can express two involutions. If $D$ is not commutative, I cannot use this technique. I wonder where these results have been. An element of group is called involution if it has order $2$.

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    $\begingroup$ This might be a naive question, but what does PSL mean for a division ring? I am confused on two counts (1) There are no determinants over non-commutative rings, so what is the difference between SL and GL? (2) The matrices of the form $c \cdot \mathrm{Id}$ are no longer a normal subgroup, so we can't quotient by them. $\endgroup$
    – David E Speyer
    Commented Oct 29, 2021 at 16:12
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    $\begingroup$ @DavidESpeyer (1) reduced norm 1 in the CSA M_2(D). (2) mod center $\endgroup$
    – Kimball
    Commented Oct 29, 2021 at 18:34

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