Every element lies in a maximal torus

Fix a bi-invariant Riemannian metric $g$ on $G$ by [L1]. Endow $G$ with the induced Riemannian distance; every closed subset of the compact space $G$ is compact by [L4], so the condition "every closed bounded subset is compact" of [L2] holds and Hopf–Rinow applies.

By [L2] there is a minimizing geodesic $\gamma :[0,1]\to G$ with $\gamma (0)=e$ and $\gamma (1)=g$; write $X:={\gamma }^{\mathrm{\prime }}(0)\in \mathfrak{g}$.

By the geodesic clause of [L1] the geodesic through the identity with initial velocity $X$ is the one-parameter subgroup $t\mapsto \exp (tX)$, so $\gamma (t)=\exp (tX)$ and in particular $g=\exp (X)$.

Let $A:=\overline{\exp (\mathbb{R}X)}$ be the closure of the image of the continuous homomorphism $t\mapsto \exp (tX)$ from the connected space $\mathbb{R}$. Then $\exp (\mathbb{R}X)$ is a connected abelian subgroup by additivity of the exponential along the line, so $A$ is a subgroup (the closure of a subgroup is a subgroup, since multiplication and inversion are continuous), it is abelian, it is closed by definition, it is compact by [L4], and it is connected by [L4] as the closure of a connected set. By [L4] it is an embedded Lie subgroup of $G$; by [L5] it is a torus, and it contains $g$ because $g=\exp (X)\in \exp (\mathbb{R}X)$.

By [L3] the torus $A$ is contained in a maximal torus $M$ of $G$, so $g\in A\subseteq M$; this includes the case of finite-order $g$, for which the same computation produces the one-parameter subgroup and no direct use of the cyclic subgroup's identity component is made. The Axiom of Choice entered through the cited averaging and structure theory.