Today we regard J. Willard Gibbs as the founder of modern, practical vector analysis. P. G. Tait, one of Hamilton's most enthusiastic followers, is quoted as writing "Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann." [Stillwell does not mention Grassmann, so let's not go there.] Well, I'm sorry, from a contemporary point of view it is Hamilton's quaternion formalism that appears to be a hermaphrodite monster, compounded of Grassmann and Gibbs (the vector space) and Lie, Klein, and Pauli (the rotational Lie algebra). [Pauli (p. 426) is the physicist who popularized the Cayley matrices as the fundamental basis for the Lie algebra of SO(3) in quantum mechanics. More accurately, Pauli's matrices differ from Cayley's (apart from the identity matrix) by a factor of i.]
The details of the correspondence between the vector space and the Lie algebra are tied up with the connection between SO(3) and another group, SU(2), comprising the rotations of 2-dimensional complex space. These groups have the same Lie algebra, abstractly, although realized as 3 x 3 and 2 x 2 matrices in the respective cases. SU(2) is twice as big as SO(3). [Technically: SO(3) is not simply connected, so it has a nontrivial universal covering, which turns out to have two sheets and to be isomorphic to SU(2). In the other direction, the mapping from SU(2) onto SO(3) is a homomorphism that is not an isomorphism.] In an angle parametrization of a curve (one-dimensional subgroup) etA in SU(2), t = 2π is not the same point as t = 0, although they map into the same rotation in SO(3). To get back where you started, you need to go on to 4π.
3-dimensional vectors can be identified with certain 2 x 2 matrices by identifying the standard basis vectors i, j, k with the corresponding Cayley matrices (see p. 426 and the exercises in Secs. 22.3 and 22.4). A rotation of the 3-vectors can be implemented by a similarity transformation x -> uxu^{-1}, where u is a matrix in SU(2). Obviously, u and -u give the same result; this is the 2-sheeted covering staring us in the face. When all the algebra and trig of this is worked out, it turns out that the most natural angle to parametrize u is half of the angle of rotation in SO(3) (reflecting identities such as sin(2t) = 2 sin t cos t). According to the article
S. L. Altmann, "Hamilton, Rodrigues, and the quaternion scandal", Math. Mag. 62 (1989) 291-308,
which I strongly recommend, Hamilton had a habit of sweeping this factor of 2 under the rug, and a more accurate understanding of what was going on was reached independently by O. Rodrigues.