Freudenthal magic square

A \ B	${\displaystyle \mathbb {R} }$	${\displaystyle \mathbb {C} }$	${\displaystyle \mathbb {H} }$	${\displaystyle \mathbb {O} }$
${\displaystyle \mathbb {R} }$	A1	A2	C3	F4
${\displaystyle \mathbb {C} }$	A2	A2 × A2	A5	E6
${\displaystyle \mathbb {H} }$	C3	A5	D6	E7
${\displaystyle \mathbb {O} }$	F4	E6	E7	E8

In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at the right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction.

The Freudenthal magic square includes all of the exceptional Lie groups apart from G₂, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G₂ itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).