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Consider we have two (general, not necessarily matrix) Lie groups $G$ and $H$. Their Cartesian product $G\times H$ is again a smooth manifold and a group, therefore a Lie group.

The vector space associated to the Lie algebra of $G\times H$ is $$ T_{(e_G,e_H)}(G\times H) = T_{e_G}G \oplus T_{e_H}=\mathfrak{g}\oplus \mathfrak{h} $$ where $\mathfrak{g}$, $\mathfrak{h}$ are the vector spaces associated to the Lie algebras of $G$ and $H$ respectively.

But what about its Lie algebra structure (i.e. Lie bracket)? How one departs from the vector space $\mathfrak{g}\oplus \mathfrak{h}$ and arrives to its Lie algebra?

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    $\begingroup$ You get the Lie bracket the same way as always. $\endgroup$ Commented Mar 21, 2020 at 20:49

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A direct sum of Lie algebras has a componentwise multiplication: $[(G_1,H_1),(G_2,H_2)]=([G_1,G_2],[H_1,H_2])$.

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