By the Riemann mapping theorem, it is also the group of conformal automorphisms of the unit disc. These Möbius transformations act as the isometries of the upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the Poincaré disk model. The above formula can be also used to define Möbius transformations of dual and double (aka split-complex) numbers. The corresponding geometries are in non-trivial relations to Lobachevskian geometry. The group SL(2,R) acts on its Lie algebra sl(2,R) by conjugation (remember that the Lie algebra elements are also 2 by 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2,R). This can alternatively be described as the action of PSL(2,R) on the space of quadratic forms on R2. The result is the following representation:  The Killing form on sl(2,R) has signature (2,1), and induces an isomorphism between PSL(2,R) and the Lorentz group SO+(2,1). This action of PSL(2,R) on Minkowski space restricts to the isometric action of PSL(2,R) on the hyperboloid model of the hyperbolic plane. The eigenvalues of an element "A" ∈ SL(2,R) satisfy the characteristic polynomial  and therefore  This leads to the following classification of elements, with corresponding action on the Euclidean plane:  The names correspond to the classification of conic sections by eccentricity: if one defines eccentricity as half the absolute value of the trace (ε = ½ tr; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2, R)), then this yields: formula_7, elliptic; formula_8, parabolic; formula_9, hyperbolic. The identity element 1 and negative identity element -1 (in PSL(2,R) they are the same), have trace ±2, and hence by this classification are parabolic elements, though they are often considered separately. The same classification is used for SL(2,C) and PSL(2,C) (Möbius transformations) and PSL(2,R) (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces; analogous classifications are used elsewhere.