= Additivity of the Lie exponential map
{title2=$\exp(X+Y)=\exp X\exp Y\iff G^\circ\text{ is abelian}$}
The <Exponential map of a Lie group> is a <group homomorphism> from the additive <Lie algebra> exactly when its <identity component of a Lie group> is abelian. If it is additive, its image is an abelian subgroup containing an identity neighborhood by the <local exponential chart>; connectedness makes that image the entire identity component. Conversely, in an abelian identity component the product $\exp(tX)\exp(tY)$ is a <one-parameter subgroup> with derivative $X+Y$, giving the identity by uniqueness. The disconnected <Lie group> $S_3\times\mathbb R$ shows that the full group need not be abelian.
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