Proof that $(1/2,1/2)$ Lorentz group representation is a 4-vector

Taken from Quantum Field Theory in a Nutshell by Zee, problem II.3.1:

Show by explicit computation that $(\frac{1}{2},\frac{1}{2})$ is indeed the Lorentz vector.

This has been asked here:

How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

but I can't really digest the formality of this answer with only a little knowledge of groups and representations.

By playing around with the Lorentz group generators it is possible to find the basis $J_{\pm i}$ that separately have the Lie algebra of $SU(2)$, and thus can be separately given spin representations.

My approach has been to write

$$J_{+i}=\frac{1}{2}(J_{i}+iK_{i})=\frac{1}{2}\sigma_{i}$$ $$J_{-i}=\frac{1}{2}(J_{i}-iK_{i})=\frac{1}{2}\sigma_{i}$$ which implies that $$J_{i}=\sigma_{i}$$ $$K_{i}=0$$ However I don't really get where to go next.