rotation of 180°. Three dimensions The Clifford algebra Cℓ3,0() is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ1, σ2 and σ3, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the pseudoscalar . It is straightforward to show that , and . The sub-algebra of even-graded elements is made up of scalar dilations, and vector rotations where corresponds to a vector rotation through an angle θ about an axis defined by a unit vector . As a special case, it is easy to see that, if , this reproduces the σ1σ2 rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ3 direction invariant, since The bivectors σ2σ3, σ3σ1 and σ1σ2 are in fact Hamilton's quaternions i, j, and k, discovered in 1843: With the identification of the even-graded elements with the algebra of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself. Thus the (real) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication. Note that the expression (1) for a vector rotation through an angle , the angle appearing in γ was halved. Thus the spinor rotation (ordinary quaternionic multiplication) will rotate the spinor through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with in place of θ/2 will produce the same vector rotation, but the negative of the spinor rotation. The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes. Explicit constructions A space of spinors can be constructed explicitly with concrete and abstract constructions. The equivalence of these constructions is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions. Component spinors Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows. Choose an orthonormal basis for V i.e. where