Graded contractions of the orthogonal Lie algebras in dimensions 7 and 8

Graded contractions of certain non-toral [Formula: see text]-gradings on the simple Lie algebras [Formula: see text] and [Formula: see text] are classified up to two notions of equivalence. In particular, there arise two large families of Lie algebras (the majority of which are solvable) of dimensions 21 and 28. This is achieved as a significant generalization of the classification of related graded contractions on [Formula: see text], the derivation algebra of the octonion algebra. Many of the results can be further extended to any subsaturated [Formula: see text]-grading on an arbitrary Lie algebra.