This article is devoted to introduce a new class of nonassociative algebras with involution including the class of structurable algebras.

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In this paper we discuss the simplicity criteria of (−1, −1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε, δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1, δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.2000 Mathematics subject classification: primary 17A40, 17B60.

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.

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In this work, we discuss a classification of [Formula: see text]-Freudenthal–Kantor triple systems defined by bilinear forms and give all examples of such triple systems. From these results, we may see a construction of some simple Lie algebras or superalgebras associated with their Freudenthal–Kantor triple systems. We also show that we can associate a complex structure into these ([Formula: see text]-Freudenthal–Kantor triple systems. Further, we introduce the concept of Dynkin diagrams associated to such [Formula: see text]-Freudenthal–Kantor triple systems and the corresponding Lie (super) algebra construction.