History · page 1 of 3

History: how \(\Esix/\Ffour\) got its shape

Five strands meet in \(\Esix/\Ffour\): a non-associative algebra, two exceptional groups, a Jordan algebra, Cartan's symmetric spaces, and Borel orbits. Each event says what was blocked, what was found, and where these pages develop it.

1843–1845 · octonions

Obstacle: After Hamilton's quaternions (1843), is there an 8-dimensional number system?
Advance: Graves found the octonions in December 1843; Cayley published them in 1845. Hamilton pointed out to Graves in July 1844 that they are not associative.
Left open: Whether non-associative numbers could support geometry or groups.
Example, picture: The Fano plane of multiplication rules, \(e_ie_{i+1}=e_{i+3}\).
Sources: [Baez, §1, p. 2], [Cayley 1845], [Graves 1845]
Developed in: quaternions and octonions (the full story, the Fano plane), Albert algebra, octonions

1887–1894 · exceptional Lie algebras

Obstacle: Classify the simple Lie algebras over \(\C\).
Advance: Killing (letter to Engel, October 1887) and Cartan (thesis, 1894) found four classical families and five exceptions, among them \(\esix\) (dim 78) and \(\ffour\) (dim 52).
Left open: What the exceptions are symmetries of.
Sources: [Baez, §4]
Developed in: the groups \(\Esix\supset\Ffour\)

1926–1927 · symmetric spaces

Obstacle: Classify Riemannian spaces with a geodesic symmetry at every point.
Advance: Cartan classified them (in modern terms, by involutions \(\theta\) of simple Lie algebras); \(\Esix/\Ffour\) is his type EIV, of rank 2.
Left open: A uniform description of the root data of all types.
Example, picture: \(\SL_3/\SO_3\) as positive-definite matrices of determinant 1.
Sources: [Cartan 1926, 1927]
Developed in: \(\SL_3/\SO_3\), the involution

1934 · the Albert algebra

Obstacle: Jordan, von Neumann and Wigner classified the algebras of quantum observables (formally real Jordan algebras), with one case left unexplained.
Advance: The exception is \(\mathfrak h_3(\Oc)\); Albert proved it is not a subalgebra of any associative algebra with \(\tfrac12(xy+yx)\).
Left open: Its symmetry groups.
Sources: [JNW], [Albert], [Baez, §3]
Developed in: Albert algebra, physics

1950 · \(\ffour\) and \(\esix\) from \(\Jalg\)

Obstacle: Realize \(\ffour\) and \(\esix\) concretely.
Advance: Chevalley and Schafer: \(\ffour=\Der(\Jalg)\) and \(\esix=\Der(\Jalg)\oplus\{\text{multiplications by traceless }Y\}\). This is the decomposition \(\kf\oplus\pf\) of the Proposition.
Sources: [CS]
Developed in: Proof of (b)

1954–1966 · the magic square

Obstacle: Explain all exceptional Lie algebras by octonions.
Advance: Freudenthal and Tits built a \(4\times4\) square of Lie algebras from pairs of division algebras. Its first row \(\so_3,\mathfrak{su}_3,\mathfrak{sp}_3,\ffour\) gives the groups \(K\) and its second row \(\mathfrak{su}_3,\mathfrak{su}_3^{\oplus2},\mathfrak{su}_6,\esix\) the groups \(G\) of the \(A_2\) row (compact forms).
Sources: [Freudenthal], [Tits], [Baez, §4.3, Table 5]
Developed in: the \(A_2\) row

1960–1962 · restricted roots and Satake diagrams

Obstacle: Read off the root data of a symmetric space from the root system of \(G\).
Advance: Satake and Araki: color the Dynkin diagram (black nodes = roots of \(\mf\)); the white nodes give the restricted root system and multiplicities.
Example, picture: EIV: four black nodes (\(D_4\)), two white ones (\(A_2\), multiplicity 8).
Sources: [Satake], [Araki]
Developed in: the Satake diagram

1974 · symmetric varieties are spherical

Obstacle: Which irreducible \(G\)-modules contain \(K\)-fixed vectors?
Advance: Vust proved that \(K^\circ P\) is open in \(G\) for suitable parabolics, giving Borel subgroups with an open orbit on \(G/K\).
Left open: The general theory of spherical varieties (Luna–Vust and later).
Sources: [Vust], [Pezzini]
Developed in: Spherical

2004–2007 · orbits over any field

Obstacle: Orbits of the norm-preserving group on \(\mathfrak h_3(C)\) over general fields, for Bhargava-style composition laws.
Advance: Krutelevich: over the split octonions the orbits are classified by rank, and elements of rank 3 with a given norm form one orbit. Over \(\C\) this is transitivity on \(\Xe\).
Sources: [Krutelevich, Prop. 12]
Developed in: Proof of (a)