If $A=B^2 = C^2$, do $B$ and $C$ commute?

Question

Let $H$ be a Hilbert space and a bounded positive operator $A$.

Let $B,C$ two operators such that $B^2 = A = C^2$. Do $B$ and $C$ commute ?

Answer 1

Here is a concrete example: $H=\mathbb C^2$, $A=I_2$, $$ B=\begin{bmatrix}0&1\\1&0\end{bmatrix},\ \ \ \ C=\begin{bmatrix}0&i\\-i&0\end{bmatrix}. $$