Exceptional Periodicity and Magic Star algebras

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but $g_2$) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call $H$ algebras) on each vertex of such a projection. We then focus on the Magic Star algebra $f_4^{\left(n\right)}$, which generalizes the non-simply laced exceptional Lie algebra $f_4$, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the $H$ algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.