Semi-simple real subal gebras of non-compact semi-simple real lie algebras V

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 when $\text{L}$ and/or $\text{L}$ is exceptional. Matrix representations for all the exceptional Lie algebras are calculated. Detailed procedures are given, which, together with those given in previous papers, allow the construction of all embeddings of $\text{L}$ in $\text{L}$, when their complex extensions are A₁, B₁, C₁, D₁, E₆, E₇, E₈, F₄, G₂ or a direct sum of any two of there. The procedures are illustrated by examples, including all real semi-simple Lie subalgebras of real forms of G₂ and sub-algebras of real forms of F₄ whose complex extensions are B₄ or A₁ (representation (16) + (9)). Because of its physical significance, all embeddings of SL (2, C) in real forms of F₄ and E₆ are given. Many of these are new results.