Does there exist an identity element in $\mathbb R^3$ with the vector cross product?

Question

Consider $\mathbb R^3$ with the vector cross product. Does this structure have an identity element? Why or why not?

Answer 1

No, such an element does not exist. Suppose toward a contradiction that $e$ is the identity element. Then $e \times v = v$ for all $v$. Plugging in $v = e$, we get that $e \times e = e$, but we have that $e \times e = 0$ by definition of the cross product, so $e = 0$. But this is absurd since $0 \times v = 0$ for all $v$, and not all vectors are the zero vector.