In mathematics, E₇ is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e₇, all of which have dimension 133; the same notation E₇ is used for the corresponding root lattice, which has rank 7. The designation E₇ comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall 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 E₇ algebra is thus one of the five exceptional cases.

The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E₇ is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56.

There is a unique complex Lie algebra of type E₇, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E₇ of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see below) of E₇, and has an outer automorphism group of order 2 generated by complex conjugation.

As well as the complex Lie group of type E₇, there are four real forms of the Lie algebra, and correspondingly four 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 133, as follows:

For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.

The compact real form of E₇ is the isometry group of the 64-dimensional exceptional compact Riemannian symmetric space EVI (in Cartan's classification). It is known informally as the "quateroctonionic projective plane" because it can be built using an algebra that is the tensor product of the quaternions and the octonions, and is also known as a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

The algebraic double cover of the complex form, or of the split real form, or of EVII can be described as the automorphism group of a Freudenthal triple system, which is a special kind of triple system defined on a 56-dimensional vector space.

The Tits–Koecher construction produces forms of the E₇ Lie algebra from Albert algebras, 27-dimensional exceptional Jordan algebras.