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)
e^{tA} 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.

- "Hamilton, Rodrigues, Gauss, quaternions, and rotations:
a historical reassessment",
*Commun. Math. Anal.***13**(2012), No. 2, 1-14. - "On Hamilton's nearly-forgotten early work on the relation
between rotations and quaternions and on the composition of rotations",
*Amer. Math. Monthly***(2014) 515-522.**