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Pair creation and annihilation in two independent oscillator modes obey $[K_{-},K_{+}]=2K_3$ and $[K_3,K_{\pm }]=\pm K_{\pm }$. The Hermitian components $K_1=(K_{+}+K_{-})/2$ and $K_2=(K_{+}-K_{-})/(2i)$ satisfy $[K_1,K_2]=-i{K}_3$. Multiplying all three Hermitian components by $i$ gives the real Lie bracket signs of SU(1,1). Both occupations change equally, so their difference is conserved. This realizes the algebra used in two-mode squeezed states.

With $J=\mathrm{diag}(1,-1)$ these matrices obey $M^{†}JM=J$ and $\det M=1$, identifying the group as SU(1,1). Products preserve the displayed conjugate-entry pattern and determinant; the identity belongs, andalso belongs. Multiplication and inversion are smooth. The real constraint $|\alpha {|}^2-|\beta {|}^2=1$ has nonzero gradient on its level set, so the matrices form a smooth real three-dimensional manifold. Associativity is inherited from matrix multiplication. These facts establish a Lie group, which is noncompact because $|\beta |$ is unbounded.