All real forms of a complex semisimple lie algebra are isomorphic

The algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is a real form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$: the $h,e,f$ form a real basis of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ and a complex basis of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, so the complex-linear extension ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}\to {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ is an isomorphism; hence ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is a real form, and it is the split real form because the adjoint operators of the real diagonal Cartan subalgebra $\mathbb{R}h$ are diagonal with real eigenvalues.

The unitary algebra $\mathfrak{s}\mathfrak{u}(2)$ is a real form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$: the map $\sigma (X)=-X^{\ast }$ is a conjugate-linear involutive automorphism of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with fixed locus $\mathfrak{s}\mathfrak{u}(2)$, and $\mathfrak{s}\mathfrak{u}(2)$ has real basis $ih$, $e-f$, $i(e+f)$.

The Killing form is negative definite on $\mathfrak{s}\mathfrak{u}(2)$ and takes a positive value on ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$: using [L2] and the basis of [L3], $B(ih,ih)=-B(h,h)=-8$, $B(e-f,e-f)=-8$, $B(i(e+f),i(e+f))=-2B(e,f)=-8$, and the mixed values $B(ih,e-f)=-iB(h,e-f)=0$, $B(ih,i(e+f))=-B(h,e+f)=0$, $B(e-f,i(e+f))=iB(e-f,e+f)=0$ vanish by [L2], so $B$ is $-8$ times a positive definite form on $\mathfrak{s}\mathfrak{u}(2)$; while $B(h,h)=8>0$ on the nonzero element $h\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$, which is therefore not a compact Lie algebra.

Isomorphisms preserve the Killing form: if $\phi \text{\hspace{0.17em}⁣}:{\mathfrak{g}}_1\to {\mathfrak{g}}_2$ is an isomorphism of real Lie algebras, then ${\mathrm{ad}}_{\phi X}=\phi {\mathrm{ad}}_X{\phi }^{-1}$, so $B_2(\phi X,\phi Y)=\mathrm{tr}({\mathrm{ad}}_{\phi X}{\mathrm{ad}}_{\phi Y})=\mathrm{tr}(\phi {\mathrm{ad}}_X{\mathrm{ad}}_Y{\phi }^{-1})=B_1(X,Y)$. Hence an isomorphism would carry the negative definite form on $\mathfrak{s}\mathfrak{u}(2)$ to the restriction of the Killing form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$, which by step 2.1 is not negative definite; consequently $\mathfrak{s}\mathfrak{u}(2)$ and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ are not isomorphic real Lie algebras.

By steps 1.1 and 1.2 both $\mathfrak{s}\mathfrak{u}(2)$ and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ are real forms of the complex semisimple Lie algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, and by step 3.1 they are not isomorphic; the statement that all real forms of a complex semisimple Lie algebra are isomorphic is therefore false.