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6.1 Definitions

Lie algebras are usually, in a first step at least, considered as complex, i.e., as complex vector spaces, structured by an antisymmetric internal bilinear product, the Lie bracket, obeying the Jacobi identity. If {T α} denotes a basis of such a complex Lie algebra 𝔤 of dimension n (over ℂ), we may also consider 𝔤 as a real vector space of double dimension 2 n (over ℝ), a basis being given by {T α, iTα }. Conversely, if 𝔤0 is a real Lie algebra, by extending the field of scalars from ℝ to ℂ, we obtain the complexification of 𝔤0, denoted by 𝔤 ℂ, defined as:
𝔤ℂ = 𝔤 ⊗ ℂ. (6.1 ) 0 ℝ
Note that (𝔤ℂ )ℝ = 𝔤0 ⊕ i𝔤0 and dimℝ (𝔤 ℂ)ℝ = 2 dimℝ (𝔤0). When a complex Lie algebra 𝔤, considered as a real algebra, has a decomposition
𝔤ℝ = 𝔤0 ⊕ i𝔤0, (6.2 )
with 𝔤0 being a real Lie algebra, we say that 𝔤0 is a real form of the complex Lie algebra 𝔤. In other words, a real form of a complex algebra exists if and only if we may choose a basis of the complex algebra such that all the structure constants become real. Note that while 𝔤ℝ is a real space, multiplication by a complex number is well defined on it since 𝔤0 ⊕ i𝔤0 = 𝔤0 ⊗ℝ ℂ. As we easily see from Equation (6.2View Equation),
ℝ ℝ ℂ × ð”¤ → 𝔤 : (a + ib,X0 + iY0) ↦→ (aX0 − bY0) + i(aY0 + bX0 ),, (6.3 )
where a,b ∈ ℝ and X0,Y0 ∈ 𝔤0.

The Killing form is defined by

B (X, Y ) = Tr (adX ad Y). (6.4 )
The Killing forms on 𝔤ℝ and 𝔤ℂ or 𝔤0 are related as follows. If we split an arbitrary generator Z of 𝔤 according to Equation (6.2View Equation) as Z = X0 + iY0, we may write:
′ ′ ′ ′ B 𝔤ℝ(Z, Z ) = 2Re B 𝔤ℂ(Z, Z ) = 2(B 𝔤0(X0, X 0) − B 𝔤0(Y0, Y0)). (6.5 )
Indeed, if ad 𝔤Z is a complex n × n matrix, ad ℝ(X0 + iY0) 𝔤 is a real 2n × 2n matrix:
(ad 𝔤 X0 − ad𝔤 Y0 ) ad𝔤ℝ(X0 + i Y0) = 0 0 . (6.6 ) ad𝔤0 Y0 ad 𝔤0 X0

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