Let $\varphi:SU(2)\times SU(2) \rightarrow End(\mathbb{H})$ given by $(A,B)\mapsto (h\mapsto AhB^{-1})$. I have shown that the kernel of this map is $\ker(\varphi)=\{(I,I),(-I,-I)\}$.
I want to calculate the derivative of $\overline{\varphi}:(SU(2)\times SU(2))/\ker(\varphi) \rightarrow End(\mathbb{H})$. But I am not sure how to do this.
Attempt: I first have tried to calculate the derivative of $\varphi$ in hopes that might help $$d_e\varphi=d/dt_t=0(h\mapsto exp(tX)hexp(-tX)$$ this gives that $d_e\varphi$ is the adjoint representation?