Euler characteristic of connected Lie group is zero or one

Statement
Suppose $$G$$ is a connected Lie group. Then $$G$$ is either a fact about::space with Euler characteristic zero or a fact about::space with Euler characteristic one. In other words, the fact about::Euler characteristic of $$G$$ is either $$0$$ or $$1$$.

The case of Euler characteristic one occurs if and only if $$G$$ is contractible. Otherwise, $$G$$ has Euler characteristic zero.

More generally, the Euler characteristic of a Lie group is equal to either zero or the number of path components.

Facts used

 * 1) uses::Euler characteristic of compact connected nontrivial Lie group is zero