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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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