The exceptional simple Lie algebras of types E7 and E8 are endowed with optimal $\mathsf{SL}_2^n$-structures, and are thus described in terms of the corresponding coordinate algebras. These are nonassociative algebras which much resemble the so-called code algebras.

Order three elements in the exceptional groups of type ${{G}_{2}}$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.

Over an algebraically closed field, there are two conjugacy classes of order three elements in ${{G}_{2}}$ in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E 8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E 8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.

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

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

Composition algebras in which the subalgebra generated by any element has dimension at most two are classified over fields of characteristic ≠2,3. They include, besides the classical unital composition algebras, some closely related algebras and all the composition algebras with invariant quadratic norm.

The action of the symmetric group $S_4$ on the tetrahedron algebra, introduced by Hartwig and Terwilliger, is studied. This action gives a grading of the algebra which is related to its decomposition into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the tetrahedron algebra and the Onsager algebra are determined.

The simple finite dimensional $(-1,-1)$ balanced Freudenthal Kantor triple systems over fields of characteristic zero are classified.

We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type $C\left( n \right),D\left( m,n \right),D\left( 2,1;\alpha\right)\left( \alpha \in \mathbb{F}\backslash \left\{ 0,-1 \right\} \right),F(4)$ , and $G(3)$ .

Alternative division quasialgebras which are not associative, are characterized. Their grading groups are shown to be always Abelian and they are built from some specific graded associative algebras by means of a graded Cayley-Dickson process.

Forms of the colour algebra introduced by Domokos and Kövesi-Domokos are studied by relating them to the well-known Cayley–Dickson algebras. Automorphisms groups and derivation algebras of these algebras are also determined.

Prime Malcev superalgebras over fields of characteristic not two and three have been studied by Shestakov [8]. He obtains the remarkable result that if these superalgebras have a nonzero odd part then they are Lie superalgebras. The main purpose of this note is to extend this result to fields of characteristic three. To this aim, it is enough to use adequately a result of Filippov [3]. Commutative and anticommutative superalgebras will be considered too, showing that they are prime, semiprime or simple as superalgebras if and only if they are as algebras. Finally, some conclusions for finite-dimensional semisimple Malcev superalgebras will be deduced. Any such superalgebra is the direct sum of a semisimple Lie superalgebra and a direct sum of simple non-Lie algebras.

The lattice of subalgebras of a Malcev algebra determines to a great extent the structureof the algebra. It is shown that conditions such as nilpotency, solvability or semisimplicity are almost characterised by means of conditions on this lattice. This enables us to study the relationship between Malcev algebras with isomorphic lattices of subalgebras.

Malcev algebras in which the relation of being an ideal is transitive are studied as well as those Malcev algebras in which every subalgebra satisfies that condition. These algebras are closely related to those in which right multiplication by any element is semisimple and they are used to determine Malcev algebras with a relatively complemented lattice of subalgebras.