For freely moving particle, It's said $L$ can't depend on velocity vector, but magnitude.

I'm looking for the contr-argument. Let's say it depends on velocity vector. Then, how would Lagrangian be written in terms of velocity vector and why would it yield the wrong solution for freely moving particle ? (no need for bringing potential in this).

Looking for the math proof. I know euler-lagrange. If it depends on velocity vector, $L = \frac{1}{2}m(v_i + v_j)$ and this yields $\frac{d}{dt}(\frac{1}{2}m) = 0$. Would this be correct approach to prove what I'm asking ? I'm trying to get the idea why Landau makes it depend on $v^2$. (I'm not asking why it doesn't depend $v^4$). My main question is to rigorously show why it can't depend on vector, and if it did, what would it break for freely moving particle ? It mentions isotrophy of space, but want to see in math proof how it's wrong.