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Cartan's method was adapted and improved for [[general relativity]] by A. Karlhede, who gave the first algorithmic description of what is now called the '''Cartan-Karlhede algorithm'''. The algorithm was soon implemented by J. Åman in an early symbolic computation engine, [[SHEEP (symbolic computation system)]], but the size of the computations proved too challenging for early computer systems to handle.
==Physical
The Cartan-Karlhede algorithm has important applications in general relativity. One reason for this is that the simpler notion of [[curvature invariant]]s fails to distinguish spacetimes as well as they distinguish [[Riemannian manifold]]s. This difference in behavior is due ultimately to the fact that spacetimes have isotropy subgroups which are subgroups of the [[Lorentz group]] SO<sup>+</sup>(3,'''R'''), which is a ''noncompact'' [[Lie group]], while four-dimensional Riemannian manifolds (i.e., with [[positive definite]] [[metric tensor]]), have isotropy groups which are subgroups of the [[compact]] Lie group SO(4).
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