## On commutators in certain Lie algebras |

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In the course of this paper it will be shown that there are many simple Lie

algebras for which this question can be answered affirmatively. All the simple Lie

algebras over the field of

classes of ...

In the course of this paper it will be shown that there are many simple Lie

algebras for which this question can be answered affirmatively. All the simple Lie

algebras over the field of

**complex numbers**are known. Among them are fourclasses of ...

Page 2

... the

as possible. It should be noted here that H. Wang and S. Pasiencier [2] have

proved that every element of a complex semi -simple 1 1 Lie group can be ...

... the

**complex numbers**and for analogues of these algebras over as many fieldsas possible. It should be noted here that H. Wang and S. Pasiencier [2] have

proved that every element of a complex semi -simple 1 1 Lie group can be ...

Page 3

It will prove necessary to make extensive use of the theory of the structure of

simple Lie algebras over the

here. Each of these simple algebras L contains a subalgebra H known as a

Cartan ...

It will prove necessary to make extensive use of the theory of the structure of

simple Lie algebras over the

**complex numbers**, a summary of which is givenhere. Each of these simple algebras L contains a subalgebra H known as a

Cartan ...

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0XXXXX 1-2 The last 2.1 is valid 2n-dimensional vector space 57 for Eg algebra of type algebras with fundamental apply exp(ad assume that Lemma automorphism exp(ad basis element bilinear form Cartan subalgebra classical Lie algebras complex numbers conjugate Cornell University corresponds a basis denoted direct sum division ring Dynkin diagram element in Eg Ellensburg Exceptional characteristics exists an automorphism exp(ad ye finite characteristic fundamental system Gordon Elliott Brown last column exists Lemma 2.X Lemma II1 Lie product linear dependence relation linearly independent mapping matrix of trace mihi+ non-singular non-zero roots positive roots preceding Lemma 3.1 Proof of Lemma prove root f simple Lie algebras simple root skew-symmetric linear transformations subspace Suppose system of roots teristic p>2 Theorem 2.1 thesis trace zero type Bn Witt algebra X 0 X XX22XXX XX2CC2X XXX2XX yields an element zero Is expressible