Why are the components of a standard d-dimensional Wiener process independent and not of a non-standard?

Question

What I mean by a standard d-dimensional Wiener process is a vector-valued stochastic process $W_t = (W^{(1)}_t, W^{(2)}_t, ..,W^{(d)}_t)$ whose components are standard one-dimensional Wiener processes. During our lecture the professor told us that these components are only independent if the components are different standard Wiener processes, hence with $\sigma^2 = 1$. Why is this not the case for Wiener processes with any $\sigma^2$?