Problems that involves two facts in the definition of vector spaces

Question

I have two small questions that I can't solve.

So, there are 8 axioms for vector spaces. One of them is that $1 \cdot a = a$ for all $a \in V$ and of course $1$ is multiplicative identity in $\mathbb{F}.$

How can I be sure this axiom cannot be derived from other 7 axioms?

Also, why is there no definition of $\alpha a= a\alpha, \forall \alpha \in \mathbb{F}, \forall a \in V$? Why is multiplication of scalars and vectors not defined in commutative way?

Answer 1

Let $V=\Bbb F^2$. For each $\alpha\in\Bbb F$ and each $(a_1,a_2)\in\Bbb F^2$, define$$\alpha.(a_1,a_2)=(\alpha a_1,0).$$Then $1.(1,1)\ne(1,1)$, but all other aximoms hold for this structure.