The Fubini Theorem for Normal Lie Subgroups of Index $2n$

Let $\Gamma_+$ be a normal subgroup of index $2n$ of a group $\Gamma$ and $\gamma_i \in \Gamma \setminus \Gamma_+$ be involutions. We first prove that if $\Gamma = \Gamma_+ \rtimes (\mathbb{Z}_2(\gamma_1) \times \cdots \times \mathbb{Z}_2(\gamma_n))$ then $\Gamma = (\Gamma_+ \rtimes \mathbb{Z}_2(\gamma_1) \rtimes \cdots \rtimes \mathbb{Z}_2(\gamma_{i-1})) \rtimes (\mathbb{Z}_2(\gamma_{i}) \times \cdots \times \mathbb{Z}_2(\gamma_n))$, where $i=2,\cdots,n$. Second, we use this result to prove the well-known Fubini theorem for a subgroup of index $2n$ of a compact Lie group. Finally, we present an application to invariant theory.