Let $G$ and $H$ be Lie groups, and let $\mathfrak g:=\operatorname{Lie}(G)$ and $\mathfrak h:=\operatorname{Lie}(H)$ be their Lie algebras. Equip $G\times H$ with the product Lie group structure. Then there is a natural [Lie algebra](/page/Lie%20Algebra) isomorphism
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\begin{align*}
\operatorname{Lie}(G\times H)\cong \mathfrak g\oplus \mathfrak h.
\end{align*}
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Under this identification, the bracket is componentwise: for all $X_1,X_2\in\mathfrak g$ and all $Y_1,Y_2\in\mathfrak h$,