Exponential of a Pauli vector ($e^{-ia\mathbf{n}\cdot \mathbf{\sigma }}=I\cos a-i\mathbf{n}\cdot \mathbf{\sigma }\sin a$)

For a real unit vector $\mathbf{n}$, the Pauli matrix multiplication law gives $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I$. Separating the even and odd powers in the matrix exponential proves the displayed identity. For spin one-half, the physical rotation through angle $\theta$ has $a=\theta /2$.

For $A=\Omega T/2$, $B=\Phi /2$, the ordered product of a $z$ rotation after an $x$ rotation has $C=\cos A\cos B$ and $\mathbf{v}=(\cos A\sin B,\sin A\sin B,\sin A\cos B)$. Since $C^2+|\mathbf{v}{|}^2=1$, its angle is $2atan2(|\mathbf{v}|,C)$ and its axis is $\mathbf{v}/|\mathbf{v}|$. At $\mathbf{v}=0$ the group element is central and its axis is undetermined.