Calculation of the Killing forms.

We shall determine first the trace bilinear form (L, M) ≡ tr LM for the standard representation of the Lie algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg104_1.tif"/> of skew linear transformations relative to a non-degenerate symmetric bilinear form (x, y) on a finite dimensional vector space https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg104_2.tif"/> . This will be used to calculate the Killing form <L, M> = tr (adL) (adM) and will serve as a starting point for deriving formulas for the Killing forms of the exceptional Lie algebras we have constructed. If u, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg104_3.tif"/> we denote the linear transformation x ⟶ (x,u)v as u ⊗ v. If v = 0, u ⊗ v = 0 and the trace tr(u ⊗ v) =0. If v ≠ 0 we choose a basis (u₁ = v, u₂,…,u_n). Then the matrix of u ⊗ v relative to this basis has (u,v) in the (1,1) position and 0’s for the other diagonal entries. Hence in all cases tr(u ⊗ v) = (u,v). It is clear from the definition of u ⊗ v that (u ⊗ v) (a ⊗ t) = (v,z) u ⊗ t. Hence (121) tr ( u ⊗ v)(z ⊗ t) = ( v , z ) ( u , t ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq270.tif"/>