= Lie product formula in a Lie group
{c}
{title2=$\exp(t(X+Y))=\lim_{m\to\infty}(\exp(tX/m)\exp(tY/m))^m$}
The <Lie product formula> extends from <matrices> to any finite-dimensional <Lie group>. In a <local exponential chart>, smooth multiplication and its differential give $\log(\exp(sX)\exp(sY))=s(X+Y)+O(s^2)$. Taking $s=t/m$ and using the <one-parameter subgroup> identity shows that the $m$th power is $\exp(t(X+Y)+O(m^{-1}))$, proving the displayed <limit>. This proof needs only the first-order expansion, rather than convergence of the full <Baker--Campbell--Hausdorff formula>.
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