Numerical Ranges Arising from Simple Lie Algebras A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized. Keywords:numerical range, convexity, inclusion relationCategories:15A60, 17B20