a compact form, for suitable trigonometric function coefficients, detailed in the Baker–Campbell–Hausdorff formula for SO(3). As a group identity, the above holds for all faithful representations, including the doublet (spinor representation), which is simpler. The same explicit formula thus follows straightforwardly through Pauli matrices; see the derivation for SU(2). For the general case, one might use Ref. Spin group The Lie group of rotation matrices, , is not simply connected, so Lie theory tells us it is a homomorphic image of a universal covering group. Often the covering group, which in this case is called the spin group denoted by , is simpler and more natural to work with. In the case of planar rotations, SO(2) is topologically a circle, . Its universal covering group, Spin(2), is isomorphic to the real line, , under addition. Whenever angles of arbitrary magnitude are used one is taking advantage of the convenience of the universal cover. Every rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2. Correspondingly, the fundamental group of is isomorphic to the integers, . In the case of spatial rotations, SO(3) is topologically equivalent to three-dimensional real projective space, . Its universal covering group, Spin(3), is isomorphic to the , . Every rotation matrix is produced by two opposite points on the sphere. Correspondingly, the fundamental group of SO(3) is isomorphic to the two-element group, . We can also describe Spin(3) as isomorphic to quaternions of unit norm under multiplication, or to certain real matrices, or to complex special unitary matrices, namely SU(2). The covering maps for the first and the last case are given by and For a detailed account of the and the quaternionic covering, see spin group SO(3). Many features of these cases are the same for higher dimensions. The coverings are all two-to-one, with , , having fundamental group . The natural setting for these groups is within a Clifford algebra. One type of action of the rotations is produced by a kind of "sandwich", denoted by . More importantly in applications to physics, the corresponding spin representation of the Lie algebra sits inside the Clifford algebra. It can be exponentiated in the usual way to give rise to a representation, also known as projective representation of the rotation group. This is the case with SO(3) and SU(2), where the representation can be viewed as an "inverse" of the