A plain dynkin diagram classifies real forms

Given: The Axiom of Choice; the complex simple Lie algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with its real forms $\mathfrak{s}\mathfrak{u}(2)=\{X:X^{\ast }=-X\}$ and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$, and the Dynkin diagram conventions of Dynkin diagram with edge multiplicity and arrow convention.

The Axiom of Choice is The Axiom of Choice; it is the hypothesis required by the Vogan-classification interface in [L4].

$\mathfrak{s}\mathfrak{u}(2)$ and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ are real forms of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$: the unitary algebra is the fixed locus of the conjugate-linear involution $X\mapsto -X^{\ast }$, and the real basis $h,e,f$ of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is a complex basis of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ (Real Cartan subalgebras need not be conjugate, The special linear Lie algebra sl_2, Classical complex matrix Lie algebras).

The Killing form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ restricts to a negative definite form on $\mathfrak{s}\mathfrak{u}(2)$ and takes the value $B(h,h)=8>0$ on the nonzero element $h=\mathrm{diag}(1,-1)\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$, so the two real forms are not isomorphic: an isomorphism preserves the Killing form, since ${\mathrm{ad}}_{\phi X}=\phi {\mathrm{ad}}_X{\phi }^{-1}$ (Killing form, Real Cartan subalgebras need not be conjugate).

The complex simple Lie algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ has Dynkin diagram $A_1$, the single-vertex diagram with no edges, and the Dynkin diagram is determined by the Cartan matrix of the root system of the complexification (Classical types correspond to sl, so and sp, Dynkin diagram with edge multiplicity and arrow convention).

The extra data beyond the plain Dynkin diagram that classify real forms are recorded by the Vogan diagram of a maximally compact Cartan subalgebra — the induced involution of the simple roots together with the painting of the fixed vertices — and equivalently by the Satake diagram of a maximally split Cartan subalgebra with its colouring and arrow pairing (Vogan diagram, Satake diagram, Classification of real forms by Vogan diagrams).