= Solution
At first order write $\alpha=1+iz$ and $\beta=x+iy$ with real $x,y,z$. The constraint forces the real part of the change in $\alpha$ to vanish. Then
$$
m=\begin{pmatrix}iz&x+iy\\x-iy&-iz\end{pmatrix}
=2x\,m_1-2y\,m_2+2z\,m_3.
$$
Thus the three supplied generators span the <tangent space> over the reals. Multiplying their matrices gives
$$
\boxed{[m_1,m_2]=m_3,\qquad[m_2,m_3]=-m_1,\qquad[m_3,m_1]=-m_2}.
$$
For example, $m_1=\sigma_x/2$, $m_2=\sigma_y/2$, $m_3=i\sigma_z/2$ makes the noncompact signs explicit.
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