Computational approach to the simplicity of <f>f4(Os,−)</f> in the characteristic two case

We present in this paper a computational approach to the study of the simplicity of the derivation Lie algebra of the quadratic Jordan algebra <f>H3(Os,−)</f>, denoted by <f>f4(Os,−)</f>, when the characteristic of the base field is two. We will show not only a collection of routines designed to find identities and construct principal ideals but also a philosophy of how to proceed studying the simplicity of a Lie algebra. We have first implemented the quadratic Jordan structure of <f>H3(Os,−)</f> into the computer system Mathematica (Computing the derivation Lie algebra of the quadratic Jordon Algebra <f>H3(Os,−)</f> at any characteristic, preprint, 2001) and then determined the generic expression of an element of the Lie algebra <f>f4(Os,−)=Der(H3(Os,−))</f> (see (41)). Once the structure of <f>f4(Os,−)</f> is completely described, it is time to analyze the simplicity by using the strategy mentioned. If the characteristic of the base field is not two, the Lie algebra is simple, but if the characteristic is two, the Lie algebra is not simple and there exists only one proper nonzero ideal <f>I</f> which is 26 dimensional and simple as a Lie algebra. In order to prove this last affirmation, we have used again the set of routines to show the simplicity of the ideal and that it is isomorphic to <f>f4/I</f>, which is also a simple Lie algebra. This isomorphism is constructed from a computed Cartan decomposition of both Lie algebras. [Copyright &y& Elsevier]