ABSTRACT
Lie algebras of type E8. Our final construction of models for the exceptional Lie algebras is one due to Tits [5] which is based on an alternative algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_1.tif"/> of degree two or one and a Jordan algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_2.tif"/> of degree three or one. Taking the parameters https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_3.tif"/> and https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_4.tif"/> to be respectively a Cayley algebra and an exceptional simple Jordan algebra one obtains forms of E8. The most important choices for https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_5.tif"/> are the composition algebras (cf. Jacobson [3]): https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_6.tif"/> a two dimensional separable associative commutative algebra, (iii) a quaternion algebra Q, (iv) a Cayley algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_7.tif"/> . The important choices for https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_8.tif"/> are: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_9.tif"/> , or a central simple Jordan algebra of degree three. According to the structure theory the possibilities one obtains here are: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_10.tif"/> , the 3 × 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_11.tif"/> -symmetric matrices with entries in https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_12.tif"/> , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_13.tif"/> or https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_14.tif"/> where in the first case https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_15.tif"/> is a central simple associative of degree three and in the second https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_16.tif"/> is simple of degree three over its center, a quadratic field over https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_17.tif"/> , and https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_18.tif"/> has an involution of second kind. Also https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_19.tif"/> denotes the set of symmetric elements of this involution. In both cases the Jordan product is a . b = 1 2 ( ab + ba ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/eq225.tif"/> . IV https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_20.tif"/> , Q a quaternion algebra, V an exceptional simple Jordan algebra. The dimensionalities for the composition algebras are respectively 1,2,4 and 8 and for the Jordan algebras listed they are 1,6,9,15,27. Taking the indicated choices for https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_21.tif"/> and https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_22.tif"/> one obtains Freudenthal’s “magic” table of Lie algebras which we shall give below. Throughout our discussion the characteristic https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756478/4cadc9cc-32ae-4840-8f83-07e1ed499955/content/pg89_23.tif"/>
