 SU(1,1)

Matrices $\left(\begin{array}{cc}\alpha & \beta \\ {\beta }^{\ast }& {\alpha }^{\ast }\end{array}\right)$ with the displayed constraint form a real three-dimensional noncompact Lie group. A real Lie algebra basis is $m_1={\sigma }_x/2$, $m_2={\sigma }_y/2$, $m_3=i{\sigma }_z/2$, with $[m_1,m_2]=m_3$, $[m_2,m_3]=-m_1$, and $[m_3,m_1]=-m_2$.