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For a vector space $V$ over a field of characteristic different from two and a quadratic form $Q$, its Clifford algebra is the quotient of the tensor algebra by the relations $v^2=Q(v)1$. Writing $B(u,v)=(Q(u+v)-Q(u)-Q(v))/2$ gives $uv+vu=2B(u,v)1$.

For a real two-dimensional positive-definite form and an orthonormal basis $e_1,e_2$, the generators satisfy $e_1^2=e_2^2=1$ and $e_1{e}_2=-e_2{e}_1$. The algebra has basis $1,e_1,e_2,e_1{e}_2$. Its Clifford modules provide spinor spaces, while products of suitable unit vectors represent rotations. The sign convention on $Q$ must be stated, especially when comparing geometric and physical conventions.