Invariant scalar product on extended Poincaré algebra

Two methods can be used to calculate explicitly the Killing form on a Lie algebra. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the group invariance of the scalar product. We use both methods in our calculation of the scalar product on the extended Poincaré algebra LG(P) in order to have a cross check of our results. The algebra is infinite-dimensional and requires careful treatment of the infinities. The scalar product on the extended algebra LG(P) found by both methods coincides and the important conclusion which follows is that Poincaré generators are orthogonal to the gauge generators. (paper)