Let $H=\sigma_3$ and $X,Y = (\sigma_1 \pm i \sigma_2)/2$ where $\sigma_k$ are the Pauli matrices. Then is there a simple algebraic way to prove that $$ S_1 \equiv \exp{ \left( \frac{\pi}{2} (X-Y)\right)} = e^X e^{-Y} e^X \equiv S_2 $$
I know 2 ways of doing this (but either not simple or not algebraic). First way is to compute the matrices directly. Second way is to show that the Adjoint action of $S_1,S_2$ on $H,X,Y$ are equal. Since $H,X,Y$ form a basis for the Lie algebra $sl(2,\mathbb{C})$ and that the center of $SL(2,\mathbb{C})$ is $\pm I$, this implies that $S_1 = \pm S_2$. However, the first way is not algebraic, while the second way seems more difficult than necessary. Is there an easier algebraic way of proving the statement?