Killing Form - Properties

Properties

• The Killing form B is bilinear and symmetric.
• The Killing form is an invariant form, in the sense that it has the 'associativity' property

B(, z) = B(x, ),

where is the Lie bracket.

• If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a scalar multiple of the Killing form.
• The Killing form is also invariant under automorphisms s of the algebra g, that is,

B(s(x), s(y)) = B(x, y)

for s in Aut(g).

• The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
• The Killing form of a nilpotent Lie algebra is identically zero.
• If I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.
• If a given Lie algebra g is a direct sum of its ideals I₁,...,I_n, then the Killing form of g is the direct sum of the Killing forms of the individual summands.