Let g be a metric on a 4 dimensional manifold M with Lorentzian signature.  The isometry group of g is the Lie group G of transformations phi : M -> M which preserves the metric g under pullback, that is, phi^*(g) = g.  Pick a point p in M.  Then the isotropy subgroup G_p of G at p is the subgroup of isometries phi in G which fix the point p, that is, phi(p) = p.  The Jacobian phi_*(p): T_pM -> T_pM defines a representation of the isotropy subgroup G_p as a subgroup of the Lorentz group SO(3, 1).  This representation is called the isotropy type of G_p.

The command IsotropyType works at the infinitesimal level.  Let Gamma be the infinitesimal isometry algebra of the metric g and let Gamma_p be the infinitesimal isotropy subalgebra of Gamma at p.  Let so(3, 1) be the Lorentz Lie algebra, viewed as a set of linear transformations on T_pM.  The mapping rho: Gamma_p -> so(3, 1) defined by rho(X)(Y) = [X, Y] (where [X, Y] is the Lie bracket of X and Y), is called the linear infinitesimal isotropy representation of Gamma_p. It gives an identification of Gamma_p with a subalgebra of the Lorentz Lie algebra so(3, 1).

The subalgebras of so(3, 1) have been classified by Patera and Winternitz and labeled as F1, F2, ..., F14. (Continuous subgroups of the fundamental groups of physics I. General method and the Poincare group, J. Math Physics, 16 (1975), 1597--1614).  Details of this classification are given in the examples below.

The command KillingVectors can be used to calculate the infinitesimal isometry algebra Gamma of the metric g.

The command IsotropySubalgebra can be used to calculate the infinitesimal isotropy subalgebra Gamma_p and the linear infinitesimal isotropy representation.

The command IsotropyType(Gamma, pt) calculates the Patera/Winternitz classification of the linear infinitesimal isotropy representation for the infinitesimal isometry group Gamma at the point p.

The calling sequence IsotropyType(output = "Notation") returns a short description of the notation used by Patera/Winternitz.  The calling sequence IsotropyType(output = "SO31I") and IsotropyType(output = "SO31II") returns the explicit matrix basis used by Patera/Winternitz for the Lie algebra of the Lorentz group.

This command is part of the DifferentialGeometry:-Tensor package and so can be used in the form IsotropyType(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-IsotropyType.

B1 = 2Rz, B2 = -2Kz, B3 = -Ry - Kz, B4 = Rx - Ky, B5 = Ry - Kx, B6 = Rx + Ky, B(theta) = cos(theta) Rz - sin(theta)Kz
F1: {B1, B2, B3, B4, B5, B6}
F2: {B1, B2, B3, B4}
F3: {Rx, Ry, Rz}
F4: {Rz, Kx, Ky}
F5: {B(theta),B3, B4}
F6: {B1, B3, B4}
F7: {B2, B3, B4}
F8: {B2 ,B3}
F9: {B1, B2}
F10: {B3, B4}
F11: {B(theta)})
F12: {Rz}
F13: {Kz}
F14: {Ry + Kz}
F15: {0}

Isotropy subalgebra has dimension 3

If isotropy subalgebra is 3 dimensional simple, then isotropy type is determined by the signature of the Killing form:
   Matrix(3, 3, [[-3/2,1/4,0],[1/4,1/8,0],[0,0,2]])
   If Killing form is negative-definite,  then isotropy type is "F3"
   If Killing form is indefinite,  then isotropy type is "F4"
   The command IsDefinite(h, 'query' = 'positive_definite') returns: false
   The command IsDefinite(h, 'query' = 'indefinite') returns: true