The identity component of a Lie group is a closed normal subgroup.

The identity component \(G_0\) of a Lie group \(G\) is a closed normal subgroup.

First, a connected component is always closed.

Second, given \(g,h \in G\), \(g\) can be seen as an homeomorphism
\[
G \longmapsto G
\]
by left multiplication. But since \(h\) and \(e\) are in the same connected component, \(gh\) and \(g\) are in the same component (and the same that \(e\)). Therefore, \(gh\in G_0\), and it is a subgroup.

And finally, since the map
\[
\phi_g:h \longmapsto ghg^{-1}
\]
is an homeomorphism of \(G\) into \(G\), and \(e\in G_0 \cap \phi_g(G_0)\), then \(\phi_g(G_0)=G_0\), so \(G_0\) is normal.

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