Split quaternion algebra

A quaternion algebra $B$ over $F$ is split if it is isomorphic as an $F$ -algebra to $M_2(F)$ . Such an isomorphism is called a splitting.

Example by matrices

For $(1,b)_F$ , take

i=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad j=\begin{pmatrix}0&b\\1&0\end{pmatrix}.

They satisfy the quaternion relations, and $1,i,j,ij$ are linearly independent when $b\ne0$ and the characteristic is not two. This gives an explicit splitting.

Change of field

Splitting depends on the field: Hamilton's real quaternion algebra is not split over $\mathbb R$ , but becomes split over $\mathbb C$ . Every quaternion algebra over an algebraically closed field of characteristic different from two splits: its presentation parameters have square roots, reducing to the matrix construction above.