TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

Appendix C
Lemma C.3. The element I always has the form:
I = α1 ∧ Ω + I1,
where I1 = 0 or I1 decomposable. More particularly, I has the following possible forms:
a) I = α1 ∧ Ω.
b) I = α1 ∧ Ω + α2 ∧ α3 ∧ (aα4 + bα5), a = 0.
c) I = α1 ∧ Ω + (cα2 + dα3) ∧ α4 ∧ α5, c = 0.
d) I = α1 ∧ Ω + α2 ∧ (aα3 − cα5) ∧ α4, a,c = 0
Proof. We rewrite I = α1 ∧Ω+α2 ∧α3 ∧(aα4 +bα5)+(cα2 +dα3)∧α4 ∧α5. If aα4 +bα5 = 0
or cα2 + dα3 = 0 then I has the form a), b) or c). If aα4 + bα5 = 0 and cα2 + dα3 = 0 then we
can assume that a = 0 and c = 0. Replace with b ′ = b a , d′ = d c and
α ′ 2 = α2 + d ′ α3, α ′ 4 = α4 + b ′ α5.
Note that our change keeps the form of Ω. Therefore, one has:
It means that I has the form d).
I = α1 ∧ Ω + aα ′ 2 ∧ α3 ∧ α ′ 4 + cα ′ 2 ∧ α ′ 4 ∧ α5.
Clearly, the forms b) and c) of I are equivalent then we only consider the form b). We rewrite
the form b) as follows:
I = α1 ∧ (pα2 ∧ α3 + qα4 ∧ α5) + α2 ∧ α3 ∧ (aα4 + bα5)
= α2 ∧ α3 ∧ (pα1 + aα4 + bα5) + qα1 ∧ α4 ∧ α5.
Replace with p ′ = p
a , b′ = b a and after that α′ 4 = p′ α1 + α4 + b ′ α5 then we obtain
For the form d), we rewirite
I = α ′ 4 ∧ (aα2 ∧ α3 − qα1 ∧ α5).
I = α2 ∧ α3 ∧ (pα1 + aα4) + (qα1 + cα2) ∧ α4 ∧ α5.
If p = q = 0 then I = α2 ∧ α3 ∧ (−aα3 + cα5) decomposable.
If p = 0, let a ′ = a p and set α′ 1 = α1 + a ′ α4 then
Let q ′ = q
c and α′ 2 = qα′ 1 + α2 then
I = pα2 ∧ α3 ∧ α ′ 1 + (qα ′ 1 + cα2) ∧ α4 ∧ α5.
I = α ′ 2 ∧ (pα3 ∧ α ′ 1 + cα4 ∧ α5).
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