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

2.2. Singular quadratic Lie algebras
If rank(C) > 2, then g is of type S1 and by Proposition 2.2.23, we have rank(C) ≥ 4.
Finally, Z(g) = CX1 ⊕ ker(C) and [g,g] = CX1 ⊕ Im(C), so g is reduced if and only if
ker(C) ⊂ Im(C).
The proof of the last claim is exactly the same as in Proposition 2.2.22 (4).
A complete classification (up to i-isomorphisms) of quadratic Lie algebras of type S3 is
given in [PU07] by applying the formula {I,I} = 0 for the case I decomposable. We shall recall
the characterization of these algebras here and describe them in terms of double extensions:
Proposition 2.2.29. Let g be a quadratic Lie algebra of type S3. Then g is i-isomorphic to an
algebra l ⊥
⊕ z where z is a central ideal of g and l is one of the following algebras:
(1) g3(λ) = o(3) equipped with the bilinear form B = λκ where κ is the Killing form and
λ ∈ C, λ = 0.
(2) g4, a 4-dimensional Lie algebra: consider q = C2 , {E1,E2} its canonical basis and the
bilinear form B defined by B(E1,E1) = B(E2,E2) = 0 and B(E1,E2) = 1. Then g4 is the
double extension of q by the skew-symmetric map
1 0
C = .
0 −1
Moreover, g4 is solvable, but it is not nilpotent. The Lie algebra g4 is known in the
literature as the diamond algebra (see for instance [Dix74]).
(3) g5, a 5-dimensional Lie algebra: consider q = C3 , {E1,E2,E3} its canonical basis and
the bilinear form B defined by B(E1,E1) = B(E3,E3) = B(E1,E2) = B(E2,E3) = 0 and
B(E1,E3) = B(E2,E2) = 1. Then g5 is the double extension of q by the skew-symmetric
map
⎛
0 1
⎞
0
C = ⎝0
0 −1⎠.
0 0 0
Moreover, g5 is nilpotent.
(4) g6, a 6-dimensional Lie algebra: consider q = C4 , {E1,E2,E3,E4} its canonical basis
and the bilinear form B defined by B(E1,E3) = B(E2,E4) = 1 and B(Ei,Ej) = 0 otherwise.
Then g6 is the double extension of q by the skew-symmetric map
⎛
0
⎜
C = ⎜0
⎝0
1
0
0
0
0
0
⎞
0
0 ⎟
0⎠
0 0 −1 0
.
Moreover, g6 is nilpotent.
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