Proving if $W(t)$ is a weiner process, then $W^2(t)$ is also a Weiner process

Question

I'm trying to solve this question:

For a stochastic process to be a Weiner Process it must have these properties:

1. $W(0) = 0$ so $W^2(0) = 0$

2. $E(W(t)) = 0$ but $E(W^2(t)) = t$ I think this is enough to prove that $W^2(t)$ is not a Weiner process

But I also want to prove the following:

3. $W^2(u) - W^2(t)$ is indepedent from $W^2(s) - W^2(r)$ given $u > t > s > r > 0$

In a Winer process the increments are normally distributed so to prove independence, I need to prove their covariance is $0$. But I don't know how to find if the increments for $W^2(t)$ are normally distributed or if there's another way to prove independence.