Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form  of g is negative-definite, otherwise  g is called non-compact.  

A Cartan decomposition is a vector space decomposition g = t 4p , where [i] t is a subalgebra, [ii] p is a subspace, [iii] [t, p] 4 p, [iv] [p, p] 4 t, [v] the Killing form is negative-definite on t and [vi] Killing form is positive-definite on p.