A real form is merely the same complex lie algebra with scalars forgotten

The realification ${\mathfrak{g}}_{\mathbb{R}}$ of a complex Lie algebra $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ is the same set with the scalar multiplication restricted to $\mathbb{R}$; it has real dimension $2{\dim }_{\mathbb{C}}\mathfrak{g}=6$, and it is closed under multiplication by $i$, that is $i{\mathfrak{g}}_{\mathbb{R}}={\mathfrak{g}}_{\mathbb{R}}$.

The unitary algebra $\mathfrak{s}\mathfrak{u}(2)=\{X\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}):X^{\ast }=-X\}$ has real dimension $3$, and it is a real form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$: the map $\sigma (X)=-X^{\ast }$ is conjugate-linear with ${\sigma }^2=\mathrm{i}\mathrm{d}$, it preserves brackets because the conjugate transpose is an anti-automorphism with $[X,Y{]}^{\ast }=-[X^{\ast },Y^{\ast }]$, and its fixed locus is exactly $\mathfrak{s}\mathfrak{u}(2)$, so by [L1] the unitary algebra is a real form.

The two real Lie algebras $\mathfrak{s}\mathfrak{u}(2)$ and ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}{)}_{\mathbb{R}}$ have different dimensions, $3$ and $6$, by steps 1.1 and 1.2, so they are not equal and not even isomorphic as real Lie algebras; and the structural difference is exactly the scalar action, since $\mathfrak{s}\mathfrak{u}(2)\cap i\mathfrak{s}\mathfrak{u}(2)=0$ while $i{\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}{)}_{\mathbb{R}}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}{)}_{\mathbb{R}}$.

The real form $\mathfrak{s}\mathfrak{u}(2)$ of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ is therefore not the underlying real Lie algebra of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$, and in general a real form ${\mathfrak{g}}_0$ of a positive-dimensional complex Lie algebra has ${\dim }_{\mathbb{R}}{\mathfrak{g}}_0={\dim }_{\mathbb{C}}\mathfrak{g}$ and ${\mathfrak{g}}_0\cap i{\mathfrak{g}}_0=0$, while the scalar-forgetful realification has twice that real dimension and is closed under multiplication by $i$; the only case in which the two can be identified is the zero algebra. What is true is the converse direction: the complexification of the real form recovers the complex algebra, ${\mathfrak{g}}_0{\otimes }_{\mathbb{R}}\mathbb{C}\cong \mathfrak{g}$, so a real form is a fixed real Lie algebra below $\mathfrak{g}$ and not $\mathfrak{g}$ with its scalars forgotten.