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Triality and Lie algebras of type $$D_4$$. (English) Zbl 0144.27103
In the present paper the author studies the forms of Lie algebras of type $$D_4$$ over an arbitrary field with the aid of Galois cohomology. For this purpose he considers two representations of the split algebra $$D_4$$ over a field $$P$$, one as the set of skew linear transformations of a split Cayley algebra $$\mathfrak C$$ over $$P$$, another as a certain set $$\mathfrak L_P$$ of transformations in $$\mathfrak C\oplus \mathfrak C\oplus \mathfrak C$$ which has close connections with triality and which is actually a sum of three inequivalent representations of the first mentioned type. The forms of $$D_4$$ over a field $$\Phi$$ which are split by a Galois extension $$P$$ of $$\Phi$$ are in 1-1 correspondence with the equivalence classes of pre-cocycles of $$G$$, the Galois group of $$P$$ over $$\Phi$$, in $$\operatorname{Aut}(\mathfrak L_P, \Phi)$$. To every such pre-cocycle corresponds a homomorphism $$p: G\to S_3$$, which plays a role in the description of the pre-cocycle. The image of $$p$$ has order $$1, 2, 3$$ or $$6$$; accordingly, the corresponding form of the Lie algebra $$D_4$$ is called of type $$D_{4\text{I}}$$, $$D_{4\text{II}}$$, $$D_{4\text{III}}$$ or $$D_{4\text{VI}}$$. A Lie algebra is shown to be of type $$D_{4\text{I}}$$ or $$D_{4\text{II}}$$ if and only if it is isomorphic with the Lie algebra of skew symmetric elements in an associative algebra with involution of type $$D_{4\text{I}}$$ or $$D_{4\text{II}}$$, respectively – cf. the author [J. Algebra 1, 288–300 (1964; Zbl 0135.07401)] or A. Weil [J. Indian Math. Soc., n. Ser. 24, 589–623 (1961; Zbl 0109.02101)].
Some more information about Lie algebras of type $$D_{4\text{I}}$$ is obtained and the automorphisms of Lie algebras of type $$D_{4\text{I}}$$ or $$D_{4\text{II}}$$ are studied. Algebras over real and $$p$$-adic fields are considered in particular.
It should be mentioned that in an appendix the author gives an important correction to a proof in a previous paper [Rend. Circ. Mat. Palermo, II. Ser. 7, 55–80 (1958; Zbl 0083.02702)].

MSC:
 17B20 Simple, semisimple, reductive (super)algebras 20J06 Cohomology of groups
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References:
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