Realification of a complex vector space by restriction of scalars

Let $W$ be a vector space over the field $\mathbb{C}$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$). Its realification, written $W_{\mathbb{R}}$, is the real vector space whose underlying set and addition are those of $W$, and whose scalar multiplication is the restriction of the complex scalar multiplication to the canonical real embedding $\mathbb{R}\subseteq \mathbb{C}$:

$r\cdot w:=rw(r\in \mathbb{R},w\in W)\mathrm{.}$

The axioms of a real vector space are the restriction of the corresponding complex axioms: complex scalar multiplication is already an $\mathbb{R}$-scalar multiplication, since $\mathbb{R}$ is a subfield of $\mathbb{C}$ through the constant-class map of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$.

Realification forgets the chosen complex structure: different complex structures on the same abelian group can have the same realification. The definition is the prototype of restriction of scalars along $\mathbb{R}\subseteq \mathbb{C}$; no basis and no choice of coordinates is involved.