What does Serge Lang mean when he says $\mathbb{R}^4=\Bbb R^3\times \mathbb{R}^1$ is equal to $\mathbb{R}^4=\mathbb{R}^2\times \mathbb{R}^2$? Serge Lang equates $\mathbb{R}^4=\mathbb{R}^3\times \mathbb{R}^1$ to $\mathbb{R}^4=\mathbb{R}^2\times \Bbb R^2$ in terms of constructing a $4$ space, but intuitively, I'm am unable to understand the first case, where he takes $x_1,x_2$ and $x_3,x_4$ separately. The part where the $3$ coordinates $x_1,x_2,x_3$ are considered separately with $x_4$ comes intuitively to me, to make a $4$ space.

He’s most likely talking about vector space isomorphisms. The isomorphism in this case would be:
$T:\mathbb{R}^3 \times \mathbb{R}^1 \to \mathbb{R}^2 \times \mathbb{R}^2 $,
$T((x_1,x_2,x_3),y) = ((x_1,x_2),(x_3,y))$.
It is easy to check this is in fact an isomorphism.
