It is instructive to compare the description of $\Sp(4,\RR)\cong\Spin(3,2)$ given in § 8.2 with the description of $\SU(2,2)\cong\Spin(4,2)$ given in § 7.5. It is clear that $\Sp(4,\RR)$ is (can be identified with) a subgroup of $\SU(2,2)$; it turns out to be the “real part” of $\SU(2,2)$. In other words, $\SU(2,2)$ turns out to deserve the name “$\Sp(4,\CC)$.” A similar construction shows that $\Sp(6,\RR)$ can be interpreted as the real part of $\SU(3,3)$, which in turn deserves the name “$\Sp(6,\CC)$.”

As discussed in § 9.4, this usage requires a choice of what one means by “symplectic”, after which the identifications above are straightforward. It should however be empahsized that this correspondence between unitary and symplectic groups fails over $\HH$ and $\OO$.

Further insight into the structure of $\Sp(6,\CC)$ can be obtained by reading § 11.5, but replacing $\OO$ everywhere by $\CC$.