Creating a linear transformation $T$ such that $T$ is invariant in a proper subspace but not normal

Question

Let $V$ be a finite-dimensional complex inner product space and let $W$ be a subspace of $V$ such that $\dim V = 4$, $\dim W = 2$. Construct a linear operator $T$ such that

(i) $W$ is $T$-invariant and $T^{*}$ invariant

(ii) $T|_{W}$ is normal

(iii) $T$ is not normal.

Does it related to 'If $W$ is $T$ invariant then $W$ is $T^{*}$ invariant' ? Thank you.