The tensor and direct-sum models of complexification are canonically complex-linearly isomorphic

The map $b:\mathbb{C}\times V\to V\oplus iV$, $b(z,v)=z(v,0)$, is $\mathbb{R}$-bilinear: it is additive in each variable because the addition in $V\oplus iV$ is componentwise, and for $r\in \mathbb{R}$ one has $b(zr,v)=(zr)(v,0)=z(r(v,0))=z(rv,0)=b(z,rv)$ by [L2].

By [L3] there is a unique $\mathbb{R}$-linear map $\mathrm{\Phi }:\mathbb{C}{\otimes }_{\mathbb{R}}V\to V\oplus iV$ with $\mathrm{\Phi }(z\otimes v)=z(v,0)$.

The map $\mathrm{\Psi }(v+iw)=1\otimes v+i\otimes w$ satisfies $\mathrm{\Psi }\circ \mathrm{\Phi }=\mathrm{id}$ on elementary tensors: writing $z=a+bi$, one has $\mathrm{\Psi }(\mathrm{\Phi }(z\otimes v))=\mathrm{\Psi }(av+ibv)=1\otimes (av)+i\otimes (bv)=a(1\otimes v)+b(i\otimes v)=(a+bi)(1\otimes v)=z\otimes v$ by the scalar action of [L1].

Conversely $\mathrm{\Phi }\circ \mathrm{\Psi }=\mathrm{id}$: $\mathrm{\Phi }(1\otimes v+i\otimes w)=(v,0)+i(w,0)=(v,0)+(0,w)=(v,w)=v+iw$ by [L2].

The map $\mathrm{\Phi }$ is complex-linear: for $z,z^{\mathrm{\prime }}\in \mathbb{C}$, $\mathrm{\Phi }(z^{\mathrm{\prime }}\cdot (z\otimes v))=\mathrm{\Phi }((z^{\mathrm{\prime }}z)\otimes v)=(z^{\mathrm{\prime }}z)(v,0)=z^{\mathrm{\prime }}(z(v,0))=z^{\mathrm{\prime }}\mathrm{\Phi }(z\otimes v)$ by [L1] and [L2], and the identity extends from elementary tensors by additivity.

Steps 3.1 and 3.2 make $\mathrm{\Phi }$ bijective with inverse $\mathrm{\Psi }$, and step 3.3 makes it complex-linear, so $\mathrm{\Phi }$ is the claimed canonical complex-linear isomorphism.