Skew-adjoint operator

In this case we say that the operator Cf is skew-adjoint. The equations... 

Unless the sets V and 0 span the same space, the operator Q is usually not self-adjoint and it may hence be described as a skew projector associated with an oblique projection of nonorthogonal character. It is immediately seen that it has the particular properties... 

Now consider an adjoint action of the Lie algebra Gc on itself, that is, examine the action of transformation of the form ad/ Gc — Gc, where h iTq. Since the element h lies in the Cart an subalgebra, it follows that the transformation adh carries into itself the plane orthogonal to the plane tTo. We make use of the fact that the operators ad are skew-symmetric with respect to the Killing form and therefore preserve the orthogonal complement by carrying it into itself. 

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