In mathematics, E₆ is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras ${\displaystyle {\mathfrak {e}}_{6}}$, all of which have dimension 78; the same notation E₆ is used for the corresponding root lattice, which has rank 6. The designation E₆ comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled A_n, B_n, C_n, D_n, and five exceptional cases labeled E₆, E₇, E₈, F₄, and G₂. The E₆ algebra is thus one of the five exceptional cases.

The fundamental group of the adjoint form of E₆ (as a complex or compact Lie group) is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. For the simply-connected form, its fundamental representation is 27-dimensional, and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional.

In particle physics, E₆ plays a role in some grand unified theories.

There is a unique complex Lie algebra of type E₆, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E₆ of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) of E₆, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.

As well as the complex Lie group of type E₆, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:

The EIV form of E₆ is the group of collineations (line-preserving transformations) of the octonionic projective plane OP². It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E₆ has a 27-dimensional complex representation. The compact real form of E₆ is the isometry group of a 32-dimensional Riemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E₇ and E₈ are known as the Rosenfeld projective planes, and are part of the Freudenthal magic square.

By means of a Chevalley basis for the Lie algebra, one can define E₆ as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") adjoint form of E₆. Over an algebraically closed field, this and its triple cover are the only forms; however, over other fields, there are often many other forms, or "twists" of E₆, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H¹(k, Aut(E₆)) which, because the Dynkin diagram of E₆ (see below) has automorphism group Z/2Z, maps to H¹(k, Z/2Z) = Hom (Gal(k), Z/2Z) with kernel H¹(k, E_6,ad).

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E₆ coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E₆ have fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); the further non-compact real Lie group forms of E₆ are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E₆ as well as the noncompact forms EI = E₆₍₆₎ and EIV = E_6(-26) are said to be inner or of type ¹E₆ meaning that their class lies in H¹(k, E_6,ad) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be outer or of type ²E₆.