An identity abut the Lie group

Question

I am studing the paper https://arxiv.org/pdf/math/0508408.pdf written by V. V. Fock and A. B. Goncharov. On page 18, I am at a loss for the identity $$E^{\alpha}H_{\alpha}(t)E^{\alpha}=H_{\alpha}(1+t^2)^{\frac{1}{2}}E^{\alpha}H_{\alpha}(1+t^{-2})^{-\frac{1}{2}},$$ where $H_{\alpha}(x)=\mathrm{exp}(\mathrm{log}(x)h_{\alpha})$, $E^{\alpha}=\mathrm{exp}e_{\alpha}$, $e_{\alpha},h_{\alpha}$ are Chevalley generators of the Lie algebra of a Lie group, $\alpha$ is a simple root. Can anyone compute it using $2\times 2$ matrices? Any help will be appreciated.