Exercise 17, § 13

Let $\mathfrak{g}$ be an absolutely simple Lie algebra of type $B_l$ (resp. $C_l,D_l$), with $l\geq 2$ (resp. $l\geq 3, l\geq 4$). Assume further that, when $\mathfrak{g}$ is of type $D_4$, the image of the homomorphism $\pi :$ Gal($\overline{k}/k$)$\rightarrow$ Aut( $\overline{R},\overline{B}$) defined in Exerc. 8 of §5 is of order $\leq 2$. Show that there exists a central simple algebra D of dimension $(2l+ 1)^2$ (resp. $4l^2,4l^2$) and an involutive anti-automorphism $\sigma$ of D such that $\mathfrak{g}$ is isomorphic to the Lie subalgebra of D consisting of the elements $x$ such that $\sigma (x) =-x$ and Tr$_D(x) = 0$ (same method as in Exerc. $16a$)$).^{22}$