Semisimple compact groups up to isogeny

Suppose ${\mathrm{\Phi }}_1\cong {\mathrm{\Phi }}_2$. By the hypothesis that the groups are semisimple, their Lie algebras and complexifications are semisimple. Match bases using the given root-system isomorphism and [L2], and in each complexification choose simple-root triples normalized for its compact conjugation ${\sigma }_i$ as in [L5]. The Cartan matrices agree, so the Serre theorem in [L5] gives a complex isomorphism $\psi :({\mathfrak{g}}_1{)}_{\mathbb{C}}\to ({\mathfrak{g}}_2{)}_{\mathbb{C}}$ sending every $e_i,f_i,h_i$ to the corresponding generator. On these generators $\psi {\sigma }_1={\sigma }_2\psi$; because they generate, the equality holds everywhere. The fixed algebra of ${\sigma }_i(U+iV)=U-iV$ is ${\mathfrak{g}}_i$, so $\psi$ restricts to a real Lie-algebra isomorphism ${\mathfrak{g}}_1\cong {\mathfrak{g}}_2$.

We construct the compact real algebra needed for realization without using later real-form theory. Let $\mathrm{\Phi }$ be a nonempty reduced crystallographic root system, choose a base with Cartan matrix $A$, and let $\mathfrak{l}=\mathfrak{g}(A)$ with simple generators $e_i,f_i,h_i$ be the Serre algebra of [L5]; [L2] identifies its root system with $\mathrm{\Phi }$. The conjugate-linear assignment $\kappa (e_i)=-f_i,\kappa (f_i)=-e_i,\kappa (h_i)=-h_i$ preserves every defining relation: it exchanges the two Serre families, exchanges the $[h_i,e_j]$ and $[h_i,f_j]$ relations, and preserves $[e_i,f_j]={\delta }_{ij}{h}_i$. It therefore descends to a conjugate-linear involutive automorphism of $\mathfrak{l}$. Its fixed algebra $\mathfrak{k}$ is a real form, since every $Z\in \mathfrak{l}$ has the unique decomposition $Z=\frac{Z+\kappa Z}{2}+i\frac{Z-\kappa Z}{2i}$ into two fixed vectors.

Let $B$ be the Killing form of $\mathfrak{l}$ and put $⟨X,Y⟩=-B(X,\kappa Y)$. On ${\mathfrak{h}}_{\mathbb{R}}={\sum }_i\mathbb{R}{h}_i$, the root decomposition gives $B(H,H)={\sum }_{\alpha \in \mathrm{\Phi }}\alpha (H{)}^2>0(H\mathrm{\ne }0),$ because the roots take real values on the $h_i$ and span ${\mathfrak{h}}^{\ast }$. Invariance gives $B(h_i,h_i)=2B(e_i,f_i)$, hence $B(e_i,f_i)>0$. Put $E_i=\mathrm{ad}{e}_i$ and $F_i=\mathrm{ad}{f}_i$. By [L9], the adjoint action of the simple-root triple is a direct sum of finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$-modules, so $E_i,F_i$ are nilpotent on each summand and $w_i=\exp (E_i)\exp (-F_i)\exp (E_i)$ is a well-defined Lie-algebra automorphism. Directly from the ${\mathfrak{s}\mathfrak{l}}_2$ brackets, $w_i$ sends $h_i$ to $-h_i$, fixes $\ker {\alpha }_i\subset \mathfrak{h}$, and sends ${\mathfrak{l}}_{\alpha }$ to ${\mathfrak{l}}_{s_i\alpha }$. On an irreducible module of highest weight $m$, use the basis $v_k=F_i^k{v}_0\mathrm{/}k!$, for which $E_i{v}_k=(m-k+1)v_{k-1}$ and $F_i{v}_k=(k+1)v_{k+1}$; the finite binomial expansion on these basis vectors gives $\exp (-F_i)\exp (E_i)\exp (-F_i)=\exp (E_i)\exp (-F_i)\exp (E_i)\mathrm{.}$ Since $\kappa E_i\kappa =-F_i$, this identity gives $\kappa w_i\kappa =w_i$. If a positive nonsimple root $\alpha =\sum n_j{\alpha }_j$ had $(\alpha ,{\alpha }_j)\le 0$ for every $j$ with $n_j>0$, then $(\alpha ,\alpha )=\sum n_j(\alpha ,{\alpha }_j)\le 0$; hence some such $j$ has $(\alpha ,{\alpha }_j)>0$, and the root-string property encoded by that finite-dimensional ${\mathfrak{s}\mathfrak{l}}_2$-module makes $s_j\alpha$ positive of smaller height. Induction, and sign change for negative roots, shows that a product of the $w_i$ carries every root to a simple root. Preservation of $B$ and commutation with $\kappa$ now give $⟨X,X⟩>0$ on every nonzero root vector, reducing by the one-dimensionality in [L9] to $⟨c{e}_i,c{e}_i⟩=\mathrm{\mid }c{\mathrm{\mid }}^{2}B(e_i,f_i)>0$. On the Cartan algebra, $⟨H,H⟩=B(H,\bar{H})>0$ by the first displayed formula. Distinct Cartan/root summands and distinct root spaces are orthogonal for this Hermitian form, because $B(\mathfrak{h},{\mathfrak{l}}_{\alpha })=0$ and $B({\mathfrak{l}}_{\alpha },{\mathfrak{l}}_{\beta })=0$ unless $\alpha +\beta =0$, while $\kappa ({\mathfrak{l}}_{\beta })={\mathfrak{l}}_{-\beta }$. Thus $⟨,⟩$ is positive definite. For $X\in \mathfrak{k}$ it equals $-B(X,X)$, so $B{\mathrm{\mid }}_{\mathfrak{k}}$ is negative definite and $\mathfrak{k}$ is compact.

For the groups in step 1.1 take the simply connected covering ${\tilde{G}}_1\to G_1$. By [L4] this covering group is compact and the covering has finite central kernel. By [L3] the real algebra isomorphism integrates to $F:{\tilde{G}}_1\to G_2$. Its differential is invertible, so local exponential charts in [L7] make F a local diffeomorphism, and its image an open subgroup; connectedness makes the image all of $G_2$. Its kernel is closed and discrete, and is central because conjugation of each kernel element gives a continuous map from connected ${\tilde{G}}_1$ into that discrete kernel. Compactness of ${\tilde{G}}_1$ makes the kernel finite. A surjective local-diffeomorphism homomorphism is a covering: choose an identity neighborhood on which it is a diffeomorphism and whose pairwise quotients meet the kernel only in the identity; its kernel translates give disjoint sheets, and translation gives the same description over every point. Thus the two maps from ${\tilde{G}}_1$ give the claimed common finite central cover.

The algebra $\mathfrak{k}$ is semisimple by [L6], and $\mathrm{Der}(\mathfrak{k})=\mathrm{ad}(\mathfrak{k})$. Every automorphism preserves its negative-definite Killing form, so $\mathrm{Aut}(\mathfrak{k})$ is the subset of the corresponding orthogonal group cut out by the finitely many closed equations $A[x,y]=[Ax,Ay]$. It is therefore compact and, by [L7], an embedded Lie subgroup. Its Lie algebra is $\mathrm{Der}(\mathfrak{k})$: differentiation gives one inclusion, while $e^{tD}$ preserves brackets for every derivation $D$. Hence $K=\mathrm{Aut}(\mathfrak{k}{)}^0$ is a compact connected Lie group with Lie algebra $\mathrm{ad}(\mathfrak{k})\cong \mathfrak{k}$. A maximal torus has a complexified Cartan subalgebra; the compact-root/Lie-root identification in [L5], followed by Cartan conjugacy in [L2], identifies its root system with that of $\mathfrak{l}$, hence with $\mathrm{\Phi }$. For the empty root system take the trivial group. This proves realization for every root system.

Conversely let $f:G_1\to G_2$ be a finite central isogeny with kernel D. By [L8], $G_1\mathrm{/}D$ is a Lie group with Lie algebra ${\mathfrak{g}}_1\mathrm{/}\mathrm{Lie}(D)={\mathfrak{g}}_1$. The induced bijection $\bar{f}:G_1\mathrm{/}D\to G_2$ is continuous, and it is a homeomorphism because its domain is compact and its target Hausdorff. Both it and its inverse are continuous homomorphisms, hence smooth by [L8]. Thus its differential is a Lie-algebra isomorphism, and so is df. A maximal toral algebra corresponds to a maximal toral algebra under this isomorphism, and [L1, L2] identify the complexified Cartan root systems. Therefore f preserves the root system, though it need not identify character lattices or root data. This also holds along a common finite central cover. Combining this converse with steps 2.2 and 2.1 proves the classification, including the trivial group.