Notes on G2: The Lie algebra and the Lie group

These notes have been prepared for the Workshop on “(Non)-existence of complex structures on S6”, celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest of the exceptional Lie groups: G2, its definition and different characterizations as well as its relationship to the spheres S6 and S7. With the exception of the summary of the Killing-Cartan classification, this survey is self-contained, and all the proofs are provided. Although these proofs are well-known, they are scattered, some of them are difficult to find, and others require stronger background, while we will usually stick to linear algebra arguments. The approach is algebraical, working at the Lie algebra level most often. We analyze the complex Lie algebra (and group) of type G2 as well as the two real Lie algebras of type G2, the split and the compact one. The octonion algebra will play its role, but it is not the starting point. Also, both the 3-forms approach and the spinorial approach are viewed and connected. Special emphasis is put on relating all the viewpoints by providing precise models.