Every unitary is a product of two-level unitaries

Let $\iota$ be a finite index set and $U(\mathbb C^\iota)$ the group of unitary $\iota\times\iota$ complex matrices. Then the subgroup generated by the two-level unitaries is the whole group:

Here $W$ is two-level if there are basis indices $a,b$ (possibly equal) such that $W_{xy}=\delta_{xy}$ whenever $x\notin\{a,b\}$ or $y\notin\{a,b\}$.

This is the first reduction in the standard proof that a gate set is universal: any unitary on $n$ qubits decomposes into unitaries that act nontrivially on only two computational basis states.