4. Given that the hyperbola with asymptotes $y= \pm 2 x$ passes through the intersection of the lines $x+y-3=0$ and $2 x-y+6=0$. Then the length of the real axis of the hyperbola is $\qquad$
$4.4 \sqrt{3}$.
It is known that the intersection point of the lines $x+y-3=0$ and $2 x-y+6=0$ is $(-1,4)$.

The equation of the hyperbola with asymptotes $y= \pm 2 x$ is $4 x^{2}-y^{2}=k$.
Solving, we get $k=-12$.
Therefore, the equation of the hyperbola is $y^{2}-4 x^{2}=12$, and its real axis length is $4 \sqrt{3}$.