Standard conjugation of a quaternion algebra

In a quaternion algebra $B=(a,b)_F$ , standard conjugation is the $F$ -linear map

\overline{x_0+x_1i+x_2j+x_3ij}=x_0-x_1i-x_2j-x_3ij.

It fixes $F$ , satisfies $\overline{\bar x}=x$ , and reverses multiplication: $\overline{xy}=\bar y\bar x$ . This is the intrinsic standard involution, independent of the chosen quaternion presentation.

Scalar coefficients stay fixed

Even when $F\subseteq\mathbb C$ , the coefficients $x_i$ are not complex-conjugated. Standard quaternion conjugation is a different operation from complex conjugation of the coefficient field.

Matrix example

In the split algebra $M_2(F)$ , it is

\begin{pmatrix}a&b\\c&d\end{pmatrix} \longmapsto\begin{pmatrix}d&-b\\-c&a\end{pmatrix}.

Multiplying a matrix by this conjugate gives its determinant times the identity.