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 $\mathrm{\φ}\:M\to M$ which preserve the metric $g$ under pullback, that is, ${\mathrm{\φ}}^{\ast }g\=g$.  Pick a point $p\in M$.  Then the isotropy subgroup $G_p\subseteq G$ at $p$ is the subgroup of isometries $\mathrm{\φ}\in G$ which fix the point $p$, that is, $\mathrm{\φ}\left(p\right)\=p$.  The Jacobian ${\mathrm{\φ}}_p^{\ast }\:T_{p}M\to T_{p}M$defines a representation of the isotropy subgroup $G_p$ as a subgroup of the Lorentz group $\mathrm{SO}\left(3\,1\right)$.  This representation is called the isotropy type of $G_p$.

The command IsotropyType works at the infinitesimal level.  Let $\mathrm{\Γ}$ be the infinitesimal isometry algebra of the metric $g$ and let ${\mathrm{\Γ}}_p$ be the infinitesimal isotropy subalgebra of $\mathrm{\Γ}$ at $p$. (${\mathrm{\Γ}}_p$ is the Lie algebra of $G_p\.\)$  Let $\mathrm{so}\left(3\,1\right)$ be the Lorentz Lie algebra, viewed as a set of linear transformations on $T_{p}M$.  The mapping $\mathrm{\ρ}\:{\mathrm{\Γ}}_p\to \mathrm{so}\left(3\,1\right)$ defined by $\mathrm{\ρ}\left(X\right)·Y \= \left[X\,Y\right]$ (where $\left[X\,Y\right]$ is the Lie bracket of $X$ and $Y$), is called the infinitesimal linear isotropy representation of ${\mathrm{\Γ}}_p$. It gives an identification of ${\mathrm{\Γ}}_p$ with a subalgebra of the Lorentz Lie algebra $\mathrm{so}\left(3\,1\right)$.

The subalgebras of $\mathrm{so}\left(3\,1\right)$ have been classified (up to conjugation) 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 IsotropyType(Gamma, p) returns the Patera-Winternitz classification of the isotropy subalgebra ${\mathrm{\Γ}}_p$.

The command KillingVectors can be used to calculate the infinitesimal isometry algebra $\mathrm{\Γ}$ of the metric $g$.

The command IsotropySubalgebra can be used to calculate the infinitesimal isotropy subalgebra ${\mathrm{\Γ}}_p$ and the linear infinitesimal isotropy representation.

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.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g1a}≔\mathrm{evalDG}\left(\frac{1}{\left(4+x^2+y^2+z^2\right)4^{-2}}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)-\mathrm{dt}\&t\mathrm{dt}\right)$

$\mathrm{g1a}:=-\mathrm{dt}\mathrm{dt}\+\frac{16\mathrm{dx}\mathrm{dx}}{4\+x^2\+y^2\+z^2}\+\frac{16\mathrm{dy}\mathrm{dy}}{4\+x^2\+y^2\+z^2}\+\frac{16\mathrm{dz}\mathrm{dz}}{4\+x^2\+y^2\+z^2}$

$\mathrm{Gamma1a}≔\mathrm{KillingVectors}\left(\mathrm{g1a}\,\mathrm{output}=''list''\right)$

$\mathrm{Gamma1a}:=\left[-\frac{1}{16}z\mathrm{D\_x}\+\frac{1}{16}x\mathrm{D\_z}\,-\frac{1}{16}z\mathrm{D\_y}\+\frac{1}{16}y\mathrm{D\_z}\,-\frac{1}{16}y\mathrm{D\_x}\+\frac{1}{16}x\mathrm{D\_y}\,-\mathrm{D\_t}\right]$

$\mathrm{IsotropyType}\left(\mathrm{Gamma1a}\,\left[x=0\,t=0\,y=0\,z=0\right]\right)$

$''F3''$

$\mathrm{IsotropyType}\left(\mathrm{Gamma1a}\,\left[x=0\,t=0\,y=0\,z=0\right]\,\mathrm{output}=''Notation2''\right)$

$''[Rx,\; Ry,\; Rz]''$

$\mathrm{DGsetup}\left(\left[t\,x\,\mathrm{\phi }\,\mathrm{\theta }\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g1b}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}-{\sinh \left(x\right)}^2\mathrm{dt}\&t\mathrm{dt}+\mathrm{dtheta}\&t\mathrm{dtheta}+{\sin \left(\mathrm{\theta }\right)}^2\mathrm{dphi}\&t\mathrm{dphi}\right)$

$\mathrm{g1b}:=-{\sinh \left(x\right)}^2\mathrm{dt}\mathrm{dt}\+\mathrm{dx}\mathrm{dx}\+{\sin \left(\mathrm{\θ}\right)}^2\mathrm{dphi}\mathrm{dphi}\+\mathrm{dtheta}\mathrm{dtheta}$

$\mathrm{Gamma1b}≔\mathrm{KillingVectors}\left(\mathrm{g1b}\right)$

$\mathrm{Gamma1b}:=\left[\frac{\cos \left(\mathrm{\θ}\right)\cos \left(\mathrm{\φ}\right)\mathrm{D\_phi}}{\sin \left(\mathrm{\θ}\right)}\+\sin \left(\mathrm{\φ}\right)\mathrm{D\_theta}\,-\frac{\cos \left(\mathrm{\θ}\right)\sin \left(\mathrm{\φ}\right)\mathrm{D\_phi}}{\sin \left(\mathrm{\θ}\right)}\+\cos \left(\mathrm{\φ}\right)\mathrm{D\_theta}\,\mathrm{D\_phi}\,-\frac{{\ⅇ}^t\cosh \left(x\right)\mathrm{D\_t}}{\sinh \left(x\right)}\+{\ⅇ}^t\mathrm{D\_x}\,\frac{{\ⅇ}^{-t}\cosh \left(x\right)\mathrm{D\_t}}{\sinh \left(x\right)}\+{\ⅇ}^{-t}\mathrm{D\_x}\,-\mathrm{D\_t}\right]$

$\mathrm{IsotropyType}\left(\mathrm{Gamma1b}\,\left[x=\mathrm{x0}\,t=\mathrm{t0}\,\mathrm{\theta }=\mathrm{\θ0}\,\mathrm{\phi }=\mathrm{\φ0}\right]\,\mathrm{output}=''Notation2''\right)$

$''[Rz,\; Kz]''$

$\mathrm{Rx},\mathrm{Ry},\mathrm{Rz},\mathrm{Kx},\mathrm{Ky},\mathrm{Kz}≔\mathrm{op}\left(\mathrm{IsotropyType}\left(\mathrm{output}=''SO31I''\right)\right)$

$\mathrm{B1},\mathrm{B2},\mathrm{B3},\mathrm{B4},\mathrm{B5},\mathrm{B6}≔\mathrm{op}\left(\mathrm{IsotropyType}\left(\mathrm{output}=''SO31II''\right)\right)$

$\mathrm{IsotropyType}\left(\mathrm{output}=''Notation''\right)$

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}

$\mathrm{IsotropyType}\left(\left[\mathrm{B1}\,\mathrm{B3}\,\mathrm{B4}\right]\right)$

$''F6''$

$\mathrm{IsotropyType}\left(\left[\mathrm{B1}+\mathrm{B3}\,\mathrm{B3}-\mathrm{B4}\,\mathrm{B1}+2\mathrm{B4}\right]\right)$

$''F6''$

$P≔\mathrm{Matrix}\left(\left[\left[1\,1\,0\,1\right]\,\left[0\,1\,1\,0\right]\,\left[1\,0\,0\,1\right]\,\left[0\,-1\,0\,1\right]\right]\right)$

$\mathrm{C1},\mathrm{C3},\mathrm{C4}≔P^{-1}·\mathrm{B1}·P,P^{-1}·\mathrm{B3}·P,P^{-1}·\mathrm{B4}·P$

$\mathrm{IsotropyType}\left(\left[\mathrm{C1}\,\mathrm{C3}\,\mathrm{C4}\right]\right)$

$''F6''$

$\mathrm{infolevel}\left[\mathrm{IsotropyType}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-IsotropyType}}:=2$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(y^3\mathrm{dt}\&t\mathrm{dt}-y^3\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-y^3\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=y^3\mathrm{dt}\mathrm{dt}-y^3\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-y^3\mathrm{dz}\mathrm{dz}$

$\mathrm{Gamma}≔\mathrm{KillingVectors}\left(g\right)$

$\mathrm{\Γ}:=\left[z\mathrm{D\_x}-x\mathrm{D\_z}\,-\mathrm{D\_z}\,-z\mathrm{D\_t}-t\mathrm{D\_z}\,-x\mathrm{D\_t}-t\mathrm{D\_x}\,-\mathrm{D\_x}\,\mathrm{D\_t}\right]$

$\mathrm{IsotropyType}\left(\mathrm{Gamma}\,\left[t=4\,x=2\,y=2\,z=1\right]\right)$

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

$''F4''$