Simple solution to the following system of equations

Question

Is there an easy way to find that $F = I$ in the following system of equations?

$$ c^2I^2-F^2v^2=c^2$$ $$ F^2-c^2J^2=1 $$ $$ c^2IJ+F^2v = 0$$ $$ c > 0 $$

Answer 1

Assume $F=I$. The system reads

$$ c^2I^2-I^2v^2=c^2,\\ I^2-c^2J^2=1,\\ c^2IJ+I^2v = 0.$$

Then from the second equation

$$J^2=\frac{I^2-1}{c^2}$$ and from the third (after squaring) $$c^4I^2\frac{I^2-1}{c^2}=I^4v^2,\\c^2(I^2-1)=I^2v^2,$$

which is compatible with the first equation.

But this works only if

$$I^2=\frac{c^2}{c^2-v^2}.$$