Compact Lie groups admit bi-invariant metrics

Choose normalized Haar measure by [L8]. By [L5] fix an inner product $(\cdot ,\cdot {)}_0$ on $\mathfrak{g}$ and define $⟨X,Y⟩:={\int }_G(\mathrm{Ad}(g)X,\mathrm{Ad}(g)Y{)}_{0}d\mu (g)$ for $X,Y\in \mathfrak{g}$; the integrand is continuous in $g$ because $\mathrm{Ad}$ is smooth and $(\cdot ,\cdot {)}_0$ is bilinear in finite dimension, so it is integrable by [L6] and the definition is unambiguous.

The form $⟨\cdot ,\cdot ⟩$ is a positive-definite inner product: it is bilinear and symmetric because the integrand is, and if $X\mathrm{\ne }0$ then $g\mapsto (\mathrm{Ad}(g)X,\mathrm{Ad}(g)X{)}_0$ is continuous, nonnegative, and equal to $(X,X{)}_0>0$ at $g=e$, so its integral is positive by [L6], while $⟨X,X⟩\ge 0$ always. It is $\mathrm{Ad}$-invariant: for $h\in G$, writing $\mathrm{Ad}(g)\mathrm{Ad}(h)=\mathrm{Ad}(gh)$ and applying the right-translation invariance [L4] to the function $g\mapsto (\mathrm{Ad}(g)X,\mathrm{Ad}(g)Y{)}_0$ gives $⟨\mathrm{Ad}(h)X,\mathrm{Ad}(h)Y⟩={\int }_G(\mathrm{Ad}(gh)X,\mathrm{Ad}(gh)Y{)}_{0}d\mu (g)=⟨X,Y⟩$.

Define the metric $g_x(u,v):=⟨d{L}_{x^{-1}}u,d{L}_{x^{-1}}v⟩$ for $x\in G$ and $u,v\in T_{x}G$; it is a smooth positive-definite bundle metric, because left translation is a diffeomorphism and $⟨\cdot ,\cdot ⟩$ is a positive-definite inner product on the single vector space $\mathfrak{g}$ by step 2.1.

The metric $g$ is right-invariant: for $h\in G$, $x\in G$ and $u,v\in T_{x}G$, write $u=d{L}_x\xi$ and $v=d{L}_x\eta$. The identity in [L3] gives $g_{xh}(d{R}_{h}u,d{R}_{h}v)=⟨{\mathrm{Ad}}_{h^{-1}}\xi ,{\mathrm{Ad}}_{h^{-1}}\eta ⟩=⟨\xi ,\eta ⟩=g_x(u,v)$ by the $\mathrm{Ad}$-invariance of step 2.1. It is left-invariant by construction, so it is bi-invariant.

For any bi-invariant metric, conjugation $C_h=L_h{R}_{h^{-1}}$ is an isometry fixing $e$, so its inner product at $e$ is $\mathrm{Ad}$-invariant. For left-invariant vector fields $X,Y,Z$ the Levi-Civita connection of $g$ satisfies ${\mathrm{\nabla }}_{X}Y=\frac{1}{2}[X,Y]$: the Koszul identity $2⟨{\mathrm{\nabla }}_{X}Y,Z⟩=X⟨Y,Z⟩+Y⟨Z,X⟩-Z⟨X,Y⟩+⟨[X,Y],Z⟩-⟨[Y,Z],X⟩+⟨[Z,X],Y⟩$, which follows from symmetry and metric compatibility in [L1], reduces to $2⟨{\mathrm{\nabla }}_{X}Y,Z⟩=⟨[X,Y],Z⟩-⟨[Y,Z],X⟩+⟨[Z,X],Y⟩$ because $X⟨Y,Z⟩=Y⟨Z,X⟩=Z⟨X,Y⟩=0$ for left-invariant fields and a left-invariant metric; differentiating this $\mathrm{Ad}$-invariance along $g=\exp (tX)$ at $t=0$ and using [L7] gives the invariance identity $⟨[X,U],V⟩=-⟨U,[X,V]⟩$, which makes the last two terms cancel and yields $⟨{\mathrm{\nabla }}_{X}Y,Z⟩=⟨\frac{1}{2}[X,Y],Z⟩$; as $Z$ ranges over a basis of $\mathfrak{g}$ and $⟨\cdot ,\cdot ⟩$ is nondegenerate, ${\mathrm{\nabla }}_{X}Y=\frac{1}{2}[X,Y]$.

Let $X\in \mathfrak{g}$ and let $\tilde{X}$ be its left-invariant field; the curve $\gamma (t):=\exp (tX)$ satisfies, by differentiating its subgroup law in [L7], ${\gamma }^{\mathrm{\prime }}(t)=\tilde{X}(\gamma (t))$, so ${\mathrm{\nabla }}_{{\gamma }^{\mathrm{\prime }}}{\gamma }^{\mathrm{\prime }}={\mathrm{\nabla }}_{\tilde{X}}\tilde{X}=\frac{1}{2}[\tilde{X},\tilde{X}]=0$ by step 5.1, and $\gamma$ is a geodesic through the identity; conversely, if $\gamma$ is a geodesic with $\gamma (0)=e$ and ${\gamma }^{\mathrm{\prime }}(0)=X$, then $\gamma$ and $t\mapsto \exp (tX)$ are geodesics with the same initial datum, so by the uniqueness in [L2] they agree wherever $\gamma$ is defined. Every such geodesic therefore extends to the displayed curve on all of $\mathbb{R}$; if maximal, its domain must be $\mathbb{R}$. Conversely each displayed global curve is maximal since no larger real interval exists. Restrictions to smaller intervals are geodesics but are not asserted to be one-parameter subgroups. This includes $X=0$, giving the constant curve, and the zero-dimensional case. AC supplies the invoked basis and Haar results and the countable choice in [L2] and [L7].