Exercise 4, § 1

(a) Suppose that 2 is invertible in $K$. Let $a$ be a Lie algebra over $K$. On the $K$-module $a' = a \times a$ we define a bracket by the formula:

Then $a'$ is a Lie $K$-algebra and the mappings $x \mapsto \frac{1}{2}(x, x)$, $x \mapsto \frac{1}{2}(x, -x)$ are isomorphisms of $a$ onto ideals $b, c$ of $a'$ of which $a'$ is the direct sum. (Consider the quadratic extension $K'$ of $K$ with basis 1 and $k$ where $k^2 = 1$. Form $a_{(K_1)}$ and then restrict the ring of scalars to $K$. Then observe that $\frac{1}{2}(1 + k)$, $\frac{1}{2}(1 - k)$ are idempotents of $K'$ and that $(1 + k)(1 - k) = 0$.)

(b) Let $a$ be a real Lie algebra, $g = a_{(\mathbf{C})}$ and $b$ the real Lie algebra derived from $g$ by restricting the field of scalars to $\mathbf{R}$. Show that $b_{(\mathbf{C})}$ is isomorphic to $a'_{(\mathbf{C})}$, where $a'$ is the algebra introduced in (a). (Consider $V = \mathbf{C} \otimes_{\mathbf{R}} \mathbf{C}$ as a vector space over $\mathbf{C}$ by $(z \otimes z')z'' = z \otimes z'z''$. Observe that $V$ is the complexification of the real vector subspace of $V$ generated by $1 \otimes 1$ and $i \otimes i$ and that this subspace is identified with the quadratic extension of $\mathbf{R}$ with basis 1 and $k$ where $k^2 = 1$.)