Kill the Killing form

Let \mathfrak g be a Lie algebra and K its Killing form. Is the kernel of K always equal to the radical of \mathfrak g?

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Kill the Killing form

No Eckmann-Hilton

Prove the following two statements and conclude that the fundamental group of a (connected) topological group G is abelian:

1) A discrete normal subgroup of a connected topological group is central.
2) If p:\hat{G}\to G is the universal covering, the total space \hat{G} admits a topological group structure such that p is a group morphism. The kernel of p is then isomorphic to the fundamental group of G.

No Eckmann-Hilton