Loop current of the loop containing a current source
It is true that the voltage between the two nodes does not depend on the current, but that does not mean it is zero. Your equation needs a term for the voltage across the current source:
$$V_{23}+I_{L2}*R_4+(R_4+R_5)I_{L1}=0$$
However, you don't know V_23, so all you've done is introduce another variable. Instead you should use the behavior of the current source to say that $$I_{L1}-I_{L3}=I_g=10mA$$ this and two loop equations will let you solve it.

Another way to do this is redraw your loops, you can leave L1 and L2 as they are, and draw L3 around the entire circuit. This way only L1 goes through the current source and you immediately know its value, leaving only two to solve for.