Decide the dimension of a Clifford algebra

Question

k is a field. The Clifford algebra $C_2(k)$ is defined as $$k\left<e_1,e_2\right> / (e_1^2+1,e_2^2+1,e_1 e_2 + e_2 e_1 = 0).$$ I know that $\{1, e_1, e_2, e_1 e_2\}$ spans $C_2(k)$. But how can I know that it is a basis?