Root diagram of exceptional lie algebra G2 and F4

Abstract

In this report, there is a compilation of many basic notations and examples from the theory of the Lie algebras. We describe some definition of General linear Lie algebra , Simple and semisimple Lie algebras , Lie groups and Lie algebra. We have now our disposal all the equipment needed to investigate the general structure of complex semisimple Lie algebras. We explain the strategy which is followed below . By using Root space decomposition, cartan-killing form and cartan matrix we describe the semisimple Lie algebra .We describe root systems and their associated Dynkin diagrams. The Cartan matrix and Dynkin diagram are introduced to suggest the applications of root systems . Finally we show how to construct the simple exceptional Lie algebra of type G2 and F4 (rank 2 and 4 respectively) and its total roots.