= Solution
A <complex analytic Lie group> is a <complex manifold> whose multiplication and inversion are <holomorphic maps between complex manifolds>. Its identity <tangent space> is a complex <Lie algebra>, and the <Adjoint representation of a Lie group>
$$
\operatorname{Ad}:G\longrightarrow\operatorname{GL}_{\mathbb C}(\mathfrak g),\qquad
\operatorname{Ad}_g=d_e(h\mapsto ghg^{-1})
$$
is holomorphic: differentiating the holomorphic conjugation map with respect to its second variable gives holomorphically varying matrix entries.
Every scalar holomorphic <function> on a compact connected <complex manifold> is constant. Indeed its modulus attains a maximum; the <maximum modulus principle>, applied in complex coordinate charts and along complex lines, makes it locally constant there, and connectedness propagates that constancy. Apply this to each entry of $\operatorname{Ad}$. Since $\operatorname{Ad}_e=I$, it follows that $\operatorname{Ad}_g=I$ for every $g$. Differentiation gives
$$
\operatorname{ad}_X(Y)=[X,Y]=0.
$$
Thus $\mathfrak g$ is abelian, and the connected <group> $G$ is abelian by the identity-neighbourhood argument in part (i). This proves that <compact connected complex Lie groups are complex tori>.
If $n=\dim_{\mathbb C}G$, identify $\mathfrak g$ with $\mathbb C^n$. The <Lie exponential map> is holomorphic, additive, onto, and a local biholomorphism, by the holomorphic left-invariant differential equation and its identity derivative. Its kernel $B$ is a <discrete subgroup> of the additive space $\mathbb C^n$. The induced map yields
$$
\boxed{G\cong\mathbb C^n/B}
$$
as complex <Lie groups>, not merely as real <groups>. Compactness further forces $B$ to be a full-rank real lattice. If its real span $W$ were proper, projection would give a surjective continuous map $\mathbb C^n/B\to\mathbb R^{2n}/W$, contradicting compactness. The discrete-subgroup argument in part (i) therefore gives $B\cong\mathbb Z^{2n}$. The quotient is a <complex torus>; it need not split into one-dimensional complex tori.
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