In this thesis, we determine the symmetric subgroups Aut(u₀)^θ for each compact simple Lie algebra u₀ and any involution θ in its automorphism group Aut(u₀). If u₀ is a classical compact simple Lie algebra, this can be done from direct calculation about matrices. If u₀ is an exceptional compact simple Lie algebra, the structure of Aut(u₀)^θ is known from the root system of its Lie algebra and corollary 2.4.3. As an application of the understanding on these symmetric subgroups, we classify Klein Four subgroups in Aut(u₀) up to conjugation in section 4.1. This leads to Marcel Berger's classification by a new approach. We also calculate the fixed point subgroups Aut(u₀)^F. After that, in section 4.2, we return to identify the conjugacy classes of involutions in symmetric subgroups. We also ca...[ Read more ]

In this thesis, we determine the symmetric subgroups Aut(u₀)^θ for each compact simple Lie algebra u₀ and any involution θ in its automorphism group Aut(u₀). If u₀ is a classical compact simple Lie algebra, this can be done from direct calculation about matrices. If u₀ is an exceptional compact simple Lie algebra, the structure of Aut(u₀)^θ is known from the root system of its Lie algebra and corollary 2.4.3. As an application of the understanding on these symmetric subgroups, we classify Klein Four subgroups in Aut(u₀) up to conjugation in section 4.1. This leads to Marcel Berger's classification by a new approach. We also calculate the fixed point subgroups Aut(u₀)^F. After that, in section 4.2, we return to identify the conjugacy classes of involutions in symmetric subgroups. We also calculate the fixed point subgroups of the Klein Four subgroups. We study the conjugation question in non-connected compact groups in section 2.5, which is used in the classification. In the last chapter, we study the fixed point subgroups Aut(so(8))^θ for order three outer automorphisms of so(8).