User:Tomruen/Freudenthal magic square

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. 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).

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