A decomposable element in the tensor product $V \otimes W$

Question

In the book of A Course in Algebra by E. B. Vinberg, at page $298$, it is given that,

An element $z \in V \otimes W$, i.e in the tensor product of the space $V$ and $W$, is called decomposable if it decomposes as $$z = x \otimes y \quad x\in V, y \in W$$

However, since $z \in V \otimes W$, by definition it has to be written as $$z = x \otimes y.$$

I mean is there any change it cannot be written in this form ? So I didn't get it what is special about decomposability of $z$.

Answer 1

Let $V=W$ be two dimensional, with basis $\{e,f\}$. Consider $$e\otimes e+f\otimes f.$$