Cartan subalgebra

Note that if desired, the Cartan subalgebra T may be interpreted as a proper subspace of transformations adp which corresponds to zero eigenvalue. The multiplicity of zero eigenvalue is equal to r, that is, to the rank of the algebra G of the dimension of the Cartan subalgebra (Fig. 14). 

In the complex case, this operator is reduced to a diagonal form in a basis composed of root vectors, whereas in the real compact case, this operator does not have real eigenvectors in the orthogonal complement of the Cartan subalgebra tTo. It is reduced to a box-diagonal form, that is, written as a matrix with (2 x 2) boxes on the diagonal. Each of these boxes corresponds to one root and is written by means of the above- mentioned matrix. 

Since the index of a Lie algebra does not change under complexification, G may be assumed to be a complex Lie algebra. Let H be the Cartan subalgebra in G such that ... 

One can construct action-angle variables on orbirts of minimal nonzero dimension in Lie algebras of the classical series Am and Cm or the Euler equations with an algebra of linear first integrals isomorphic to a Cartan subalgebra. After this, the Euler equations are integrated in elementary functions. 

As the Cartan subalgebra H in the algebra so(3) consider the linear subspace generated by an element 623 so(3). Suppose that... 

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