The 8 axioms of holding for a vector space defined in Chapter 5 of the book Robertson's Basic Linear Algebra are easily checked for tensor product expect for existence of (unique?) additive inverse and identity:
(V3) there exists an element $0 \in V$ such that $x + 0 = x$ for every $x \in V$.
(V4) for every $x \in V$ there exists $-x \in V$ such that $x + (-x) = 0$;
For example, the problem is that sum is defined when at most one component differs. So in order $(v,w) + 0 = (v,w)$ to hold, then $0=(0,w)$ or $0=(v,0)$ not only not defining a unique $0$ but also it implies $(0,w)=(v,0)$. ...
How to check the mentioned axioms of a vector space, i.e. (V3) and (V4), for the tensor-product vector space?
PS I am learning tensor product from this link and I because it claimed that tensor product is a vector space "by force [=definition] I tired to check that by what I had learnt from Robertson's Basic Linear Algebra.