Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\top \mathbf{x}$ is independent of $(\mathbf{I}_n-\mathbf{u}\mathbf{u}^\top)\mathbf{x}$ and trivially their sum is $\mathbf{x}$ itself.

My question is as following. Let $\mathbf{y}$ be a random vector sampled from a uniform coordinate distribution, that is $\mathbf{y}$ is distributed uniformly across $\{\sqrt{n}\mathbf{e}_1, \sqrt{n}\mathbf{e}_2,\cdots,\sqrt{n} \mathbf{e}_n\}$, where $\mathbf{e}_i$ is the $i$-th standard basis vector.

Can we similarly decompose $\mathbf{y}$ into two parts such that those two parts are independent of each other?