All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six.

The maximal solvable subalgebras of the real classical Lie algebrasThe Lie subalgebras of all real Lie algebras of dimension d =:::; 4 are classified into equivalence classes under their groups of inner automorphisms. Tables of representatives of each conjugacy class are given.{e3}, {eJ. {el+xe3} {el He,J. {e,He,J, {e,HeJ {el + xe4}, {e,J. {e, cos1> + e3 sin¢ + xe,J {el}. {et+Ee,J. {e,+xe,J, {e,+ xe4}, {e4} {e, cos1> + e, sin1>} , {e3 + xe,J. {eJ, {et cos¢ + e, sin¢ + f eJ jet}, {e,}. {e,J, {e,+xe,J, {el + fe, + xe,J, {e, +E eJ, {e,+E e4) {el}, {e 2 L {e,J, {et +fell {e2+ feJ, {e,+ xe,J, {e! + £e,+ xe,J {eJ. {et + xe4}' {e3 + yeJ (x>-0) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded

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Three types of numerical data are provided for compact simple Lie groups G of classical types and of any rank. This data is indispensable for Fourier-like expansions of multidimensional digital data into finite series of E−functions on the fundamental domain F e . Firstly, we determine the number |F e M | of points in F e from the lattice P ∨ M , which is the refinement of the dual weight lattice P ∨ of G by a positive integer M . Secondly, we find the lowest set Λ e M of the weights, specifying the maximal set of E−functions that are pairwise orthogonal on the point set F e M . Finally, we describe an efficient algorithm for finding the number of conjugate points to every point of F e M . Discrete E−transform, together with its continuous interpolation, is presented in full generality.

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