We know that when $n$ is odd, $\operatorname{O}_n(\mathbb R) \simeq \operatorname{SO}_n (\mathbb R) \times \mathbb Z_2$.

However, this seems **not true when $n$ is even**. But I have no idea how to prove something is not a direct product.

I have tried to verify some basic properties of direct product. For example, $\operatorname{SO}_n(\mathbb R)$ is a normal subgroup of $\operatorname{O}_n(\mathbb R)$, whenever $n$ is odd or even. But they are not helpful.

So, is this statement true and how to prove it?

Thank you!