Properties

The Killing anatomy B is bilinear and symmetric.

The Killing anatomy is an invariant form, in the faculty that it has the 'associativity' property

B([x,y],z)=B(x,[y,z]),

area [,] is the Lie bracket.

If g is a simple Lie algebra again any invariant symmetric bilinear anatomy on g is a scalar assorted of the Killing form.

The Killing anatomy is additionally invariant beneath automorphisms s of the algebra g, that is,

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

for s in Aut(g).

The Cartan archetype states that a Lie algebra is semisimple if and alone if the Killing anatomy is non-degenerate.

The Killing anatomy of a nilpotent Lie algebra is analogously zero.

If I and J are two ethics in a Lie algebra g with aught intersection, again I and J are erect subspaces with account to the Killing form.

If a accustomed Lie algebra g is a absolute sum of its ethics I1,...,In, again the Killing anatomy of g is the absolute sum of the Killing forms of the alone summands.