In particular, a real Lie algebra \ mathfrak g is called "'compact "'if the Killing form is negative definite.
12.
The first one is noncompact, the so-called "'split real form "', and its Killing form has signature.
13.
This set of roots form a root system inside \ mathfrak { h } ^ *, as defined above, where the inner product is the Killing form.
14.
Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,.
15.
The special feature of a Cartan decomposition is that the Killing form is negative definite on \ mathfrak { k } and positive definite on \ mathfrak { p }.
16.
When the Killing form of the Lie algebra is contracted with the current commutator, one obtains the energy-momentum tensor of a two-dimensional conformal field theory.
17.
If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero ( and more generally the Killing form vanishes on any nilpotent ideal ).
18.
If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero ( and more generally the Killing form vanishes on any nilpotent ideal ).
19.
By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries + 1 or " 1.
20.
Furthermore, \ mathfrak { k } and \ mathfrak { p } are orthogonal complements of each other with respect to the Killing form on \ mathfrak { g }.
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