product results in If is identified with the pseudoscalar then the right hand side becomes which is also the definition for the product of two vectors in geometric algebra. If we define the spin operator as , then satisfies the commutation relation:Or equivalently, the Pauli vector satisfies: Some trace relations The following traces can be derived using the commutation and anticommutation relations. If the matrix is also considered, these relationships become where Greek indices and assume values from and the notation is used to denote the sum over the cyclic permutation of the included indices. Exponential of a Pauli vector For one has, for even powers, which can be shown first for the case using the anticommutation relations. For convenience, the case is taken to be by convention. For odd powers, Matrix exponentiating, and using the Taylor series for sine and cosine, . In the last line, the first sum is the cosine, while the second sum is the sine; so, finally, which is analogous to Euler's formula, extended to quaternions. Note that , while the determinant of the exponential itself is just , which makes it the generic group element of SU(2). A more abstract version of formula for a general matrix can be found in the article on matrix exponentials. A general version of for an analytic (at a and −a) function is provided by application of Sylvester's formula, The group composition law of A straightforward application of formula provides a parameterization of the composition law of the group . One may directly solve for in which specifies the generic group multiplication, where, manifestly, the spherical law of cosines. Given , then, Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to (Of course, when is parallel to , so is , and .) Adjoint action It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle along any axis : Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that . Completeness relation An alternative notation that is commonly used for the Pauli matrices is to write the vector index in the superscript, and the matrix indices as subscripts, so that the element in row