Exceptional and semi simple Lie groups hierarchies and the maximum number of elementary particles beyond the standard model of high energy physics

In this work and following some major developments in our understanding of the connection between the exceptional Lie symmetry groups hierarchy and physics, we present finite and infinite dimensional Lie groups and their connections to the orthogonal and unitary groups. The dimension of the Lie group is included in the derivation of the number of elementary particle-like super states 576 and 672. Connections to the dimensions of Lie groups and the 72 and 84 elementary particles recently considered by the Author are discussed.