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Suppose I have a group homomorphism $\rho:SL(2,\mathbb{C})\to SO_0(3,1)$ given by $\rho(a)X=aXa^*$ and I want to see how the corresponding Lie map $L\rho$ looks like. By definition $$ L\rho=\frac{\mathrm{d}}{\mathrm{d}t}\rho(\exp (tX))Y\bigg|_{t=0} $$ which is supposed be equal to $XY+YX^*$, but I fail to see how.

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Hint: Use the product rule to differentiate $\exp(tX) Y \exp(tX)^*$. – Michael Joyce Mar 30 '13 at 18:42

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