Example

The matrix algebra $\mathbf{M}_2(K)$ is isomorphic to the quaternion algebra of type $(0,1,1)$. Indeed, consider the quadratic algebra $E = K\times K$ (of type $(0,1)$) and the quaternion algebra $F = E + Ej$, which is the Cayley extension of E defined by the element $\gamma = 1$. The mapping $(a, b)\mapsto \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ is an algebra homomorphism from E to $\mathbf{M}_2(K)$. Since for $a, b$ in K, we have

=, =,

1 0 1 0 0 1 1 0 0 $b$ 0 $a$ 1 0

this homomorphism extends to an algebra homomorphism $\theta : F\longrightarrow \mathbf{M}_2(K)$ defined by

This homomorphism is bijective. When the characteristic of K is different from 2, the algebra $\mathbf{M}_2(K)$ is also isomorphic to the quaternion algebra of type $(1,0,1)$ (Remark 1).