I wish to prove the following identity $$ \Big(\vec{a}\cdot\vec{\sigma}\Big) \left(\vec{b}\cdot\vec{\sigma}\right) = \left(\vec{a}\cdot\vec{b}\right)I+i \left(\vec{a}\times\vec{b}\right) \cdot \vec{\sigma} $$
regarding pauli matrices and two arbitrary vector operators. The identity's proof is given in Wikipedia, and is very straightforward.
However, it relies on the assumption that the element (matrices) of the vector operator $\vec{b}$ commute with the Pauli matrices.
My question is: Is the identity only valid under this assumption or am I missing something?