On the decomposition of the algebra of the group of arbitrary four-dimensional quadratic form

The six-dimensional algebra of the group of motions of arbitrary homogenenous four-dimensional quadratic form is considered. It is shown that this algebra can be decomposed into the product of two subalgebras by means of an arbitrary constant four-dimensional vector or two niplotent operators representing the constant antisymmetric mixed (doubly contravariant and doubly covariant) tensors of fourth rank in four-dimensional space. Some consequences of such decomposition are discussed in connection with the fact that the special cases of the group of motions of arbitrary homogeneous four-dimensional quadric are the group of four-dimensional rotations SO(4) and the Lorentz group SO(1,3). 10 refs