Say there are two vectors, $v_1$ and $v_2$ that come from a set of vectors $V$
The definition of the commutative property is (coming from one of the eight axioms for a vector space): $v_1 + v_2 = v_2 + v_1$
However, is it also necessary for $v_1 + v_2$ and $v_2 + v_1$ to be elements of the set V to consider $v_1$ and $v_2$ commutative? Or does a set have to be specified when mentioning the commutative property, e.g. "commutative over $R^2$ but not necessarily $V$?
Note that "+" is a function mapping elements of $V \times V$ to $V$. Hence it is defined for all pairs of elements $(x,y)$ of $V$. In particular, both $x+y =: +(x,y) \in V$ and $+(y,x) \in V$, since $V$ is the space "+" maps into by assumption. In other words, $v_1 + v_2$ and $v_2 + v_1$ are always elements of $V$, because of how we define "+".