  We expand on an idea of Vinberg to take a tensor space and the natural Lie
algebra that acts on it and embed their direct sum into an auxiliary algebra.
Viewed as endomorphisms of this algebra, we associate adjoint operators to
tensors. We show that the group actions on the tensor space and on the adjoint
operators are consistent, which means that the invariants of the adjoint
operator of a tensor, such as the Jordan decomposition, are invariants of the
tensor. We show that there is an essentially unique algebra structure that
preserves the tensor structure and has a meaningful Jordan decomposition. We
utilize aspects of these adjoint operators to study orbit separation and
classification in examples relevant to tensor decomposition and quantum
information.
