  A $G$-grading on an algebra is called multiplicity free if each homogeneous
component of the grading is 1-dimensional, where $G$ is an abelian group. We
introduce skew root systems of Lie type and skew root systems of Jordan type
respectively, and use them to construct multiplicity free gradings on
semisimple Lie algebras and on semisimple Jordan algebras respectively. Under
certain conditions the corresponding Lie (resp. Jordan) algebras are simple.
Three families of skew root systems of Lie type (resp. of Jordan type) are
constructed and the corresponding Lie (resp. Jordan) algebras are identified.
This is a new approach to study abelian group gradings on Lie and Jordan
algebras.
