# Proof regarding affine spaces over fields

Consider an affine space $$\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$$, with $$\mathbb{K}$$ a field. Show that $$\mathcal{A}$$ contains $$\operatorname{AG}(2,2)$$ if and only if $$\operatorname{char}(\mathbb{K})=2.$$

I have no idea how to start with this. I still tried something:

$$\Rightarrow$$: suppose $$\operatorname{AG}(2,2) \subseteq \mathcal{A}$$. This means that $$\mathcal{A}$$ contains $$4$$ different points $$A,B,C,D$$ of which no $$3$$ are collinear and $$AB \parallel CD$$, $$AC \parallel BD$$, $$AD \parallel BC$$. Now I'm stuck. I believe I'll have to introduce a multiplicative identity $$e$$ and show that $$e+e = 0$$.

$$\Leftarrow$$: suppose $$\operatorname{char}(\mathbb{K}) = 2$$. This means that $$|\mathbb{K}| = 2^h$$, with $$h\ge 1$$ because a line has at least $$2$$ points. Somehow I'll have to prove that $$h=2$$.

Can someone help? Thanks.