Exercise 3, § 13

a) Define an isomorphism from $\mathfrak{s}\mathfrak{l}(4, k)$ to the orthogonal algebra $\mathfrak{o}_S(6, k)$. (Use the fact that the representation $\bigwedge^2\sigma$ of no. 1.V is orthogonal and of dimension 6.) The two types of irreducible representation of $\mathfrak{s}\mathfrak{l}(4, k)$ of degree 4 correspond to the two semi-spinor representations of $\mathfrak{o}_S(6, k)$.

b) Define an isomorphism from $\mathfrak{s}\mathfrak{p}(4, k)$ to the orthogonal algebra $\mathfrak{o}_S(5, k)$. (Use the fact that the representation $\sigma_2$ of no. 3.V is orthogonal and of dimension 5.) The irreducible representation of $\mathfrak{s}\mathfrak{p}(4, k)$ of degree 4 corresponds to the spinor representation of $\mathfrak{o}_S(5, k)$.

c) Define an isomorphism $\mathfrak{s}\mathfrak{l}(2, k)\times \mathfrak{s}\mathfrak{l}(2, k)\rightarrow \mathfrak{o}_S(4, k)$ by using the tensor product of the identity representations of the two $\mathfrak{s}\mathfrak{l}(2, k)$ factors. Recover this result by means of Chap. I, §6, Exerc. 26.