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 $z$.