Is every plane in $\mathbb{R}^4$ a line in $\mathbb{C}^2$?

Question

Every complex line, that is, one-dimensional complex affine space, in $\mathbb{C}^2$ is a real plane in $\mathbb{R}^4$. Is the converse true? That is, is every real plane in $\mathbb{R}^4$ a complex line in $\mathbb{C}^2$?

Answer 1

Consider the set $$ A=\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4{\,\mid\,}x_1=x_2\;\text{and}\;x_3=x_4\} $$ Then $A$ is a real plane through the origin in $\mathbb{R}^4$.

Suppose $A$ is also a complex line through the origin in $\mathbb{C}^2$ with the identification $$(x_1,x_2,x_3,x_4)\;{\small{\longleftrightarrow}}\;(x_1+x_2i,x_3+x_4i)=(z_1,z_2)$$

Then for some $a,b\in\mathbb{C}$, not both zero, we would have $$ A=\{(z_1,z_2)\in\mathbb{C}^2{\,\mid\,}az_1+bz_2=0\} $$ But then from $(0,0,1,1)\in A$ in $\mathbb{R}^4$ we would get $b=0$, and from $(1,1,0,0)\in A$ in $\mathbb{R}^4$ we would get $a=0$, contradiction.