Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$′ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of $\text{L}$′ in $\text{L}$ when their complex extensions are A₁, B₁, C₁, D₁ or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A₁?(A₂⊕A₂), (A₁⊕A₁)?(A₂⊕A₂), (A₁⊕A₁)?A₃, (A₁⊕A₁)??(A₃⊕A₃) and (A₁⊕A₁)?(A₃⊕A₂). Because of its physical significanc embeddings of SL(2,C) in simple and semi-simple real Lie algebras are studied in detail.