All the abstract ten-dimensional real Lie algebras that contain as a subalgebra the algebra of the three-dimensional rotation group (generators J) and decompose under the rotation group into three three-vector representation spaces (J itself, K, and P) and a scalar (generator H) are classified. In all cases, the existence of a homogeneous space of dimension 4 is shown.
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Bacry et al. (1986) studied this question.
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