existence and uniqueness of compact real form

Let $G$ be a semisimple complex Lie group. Then there exists a unique (up to isomorphism) real Lie group $K$ such that $K$ is compact and a real form of $G$. Conversely, if $K$ is compact, semisimple and real, it is the real form of a unique semisimple complex Lie group $G$. The group $K$ can be realized as the set of fixed points of a special involution of $G$, called the Cartan involution.

For example, the compact real form of ${\mathrm{SL}}_nℂ$, the complex special linear group, is $\mathrm{SU}(n)$, the special unitary group. Note that ${\mathrm{SL}}_nℝ$ is also a real form of ${\mathrm{SL}}_nℂ$, but is not compact.

The compact real form of ${\mathrm{SO}}_nℂ$, the complex special orthogonal group, is ${\mathrm{SO}}_nℝ$, the real orthogonal group. ${\mathrm{SO}}_nℂ$ also has other, non-compact real forms, called the pseudo-orthogonal groups.

The compact real form of ${\mathrm{Sp}}_{2n}ℂ$, the complex symplectic group, is less well-known. It is (unfortunately) also usually denoted $\mathrm{Sp}(2n)$, and consists of $n\times n$ “unitary” quaternion matrices, that is,

$$\mathrm{Sp}(2n)=\{M\in {\mathrm{GL}}_nℍ|M{M}^{*}=I\}$$

where $M^{*}$ denotes $M$ conjugate transpose. This different from the real symplectic group ${\mathrm{Sp}}_{2n}ℝ$.