Example

(1) Let $A$ be a quadratic algebra over $K$ of type $(\alpha, \beta)$ and $(e_1, e_2)$ a basis of type $(\alpha, \beta)$ (§ 2, no. 3). For $x = \xi e_1 + \eta e_2$, $\mathrm{Tr}_{A/K}(x) = 2\xi + \beta \eta$ and $N_{A/K}(x) = \xi^2 + \beta \xi \eta - \alpha \eta^2$; these functions are therefore identical with the Cayley trace and norm of $x$ (§ 2, no. 24).

(2) Let $A$ be a quaternion algebra over $K$. A direct calculation allows us to verify that $\mathrm{Tr}_{A/\mathbf{K}}(x) = 2T(x)$ and $N_{A/\mathbf{K}}(x) = (N(x))^2$, where T and N are the Cayley trace and norm (§ 2, no. 4).

(3) Let $A = \mathbf{M}_n(\mathbf{K})$ and let the canonical basis $(E_{ij})$ of A (II, § 10, no. 3) be arranged in lexicographic order. Then it is immediately seen that for every matrix $X = \sum_{i,j} \xi_{ij} E_{ij}$ the matrix (of order $n^2$) of the endomorphism $Y \mapsto XY$ is of the form

whence $\mathrm{Tr}_{A/\mathbf{K}}(X) = n \cdot \mathrm{Tr}(X)$ and $N_{A/\mathbf{K}}(X) = (\det(X))^n$.