Special Freudenthal-Kantor triple systems and Lie algebras with dicyclic symmetry

Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3 are considered. The close connections of these algebras with Lie algebras graded over the nonreduced root system BC1, with J-ternary algebras and with Freudenthal-Kantor triple systems are explored.