Does it mean that a given rotation in 4-dimensional Euclidean space cannot be associated with a unique axis ($\hat{\textbf{n}}$) of rotation? If yes, why is that the case?
Yes, this is absolutely true. The notion of a one dimensional axis is an "accident" of three dimensions. Rotations transform planar (dimension 2) linear subspaces of Euclidean space and so one needs to specify the transformed plane and the rotation angle to specify the rotation.
In 3D dimensions we can cheat a little: a plane is uniquely defined by a unit normal vector, and the rotation angle can be encoded as the length of this vector. This is what we mean by an axis. The axis is the untransformed space of the rotation; the 3D space splits into two orthogonal, invariant spaces, the former being the plane of rotation, which is invariant but transformed (i.e. nontrivially bijectively mapped to itself) and the latter the axis, which is both invariant and untransformed. In 4 and higher dimensions, the invariant spaces are of 2 or higher dimensions.
A member of the Lie algebra of a rotation group (with the algebra written as a faithful matrix representation) is a skew-symmetric matrix, i.e. an entity of the form $\sum\limits_i X_i \wedge Y_i$ where the $X_i$ and $Y_i$ are 1D vectors in the Euclidean space. A general rotation matrix is then of the form $\exp\left(\sum\limits_i X_i \wedge Y_i\right)$. Things get kind of complicated in 4 and higher dimensions; the most general thing one can say is that a general proper orthogonal transformation on $N$ dimensional space can be decomposed as $R_1\circ R_2\circ\,\cdots R_{N\,\mathrm{div}\, 2}$ where each of the $R_i$ is a rotation that bijectively transforms a plane into itself and leaves the plane's complement invariant. However, the planes for each of the $R_i$ are not in general the same plane.
Further Questions and Useful Rotation Properties
User John Dvorak points out:
I would think that $R_1\circ R_2\circ\,\cdots R_{N\,\mathrm{div}\, 2}$ would always be pairwise orthogonal. Is that not the case?
This is indeed absolutely true and it is worth sketching the proof to get more insight into a higher dimensional rotation.
Let our rotation matrix be $R=\exp(H)$ with $H=\sum\limits_i X_i \wedge Y_i\in \mathfrak{so}(N)$ as above. Then there exists another orthogonal transformation $\tilde{R}$ (i.e. $\tilde{R}\in \mathrm{SO}(N)$) that, through similarity transformation, reduces the skew symmetric $H\in \mathfrak{so}(N)$ to block diagonal form:
$$H = \tilde{R}\,\mathrm{diag}(\Lambda_1,\,\Lambda_2,\,\cdots)\,\tilde{R}^T=\tilde{R}\,\mathrm{diag}(\Lambda_1,\,\Lambda_2,\,\cdots)\,\tilde{R}^{-1}$$
where each of the blocks is of the form:
$$\Lambda_j=\left(\begin{array}{cc}0&-\theta_j\\\theta_j&0\end{array}\right)$$
with $\theta_j\in\mathbb{R}$ being a rotation angle and that, if $N$ is odd, there is also a $1\times1$ zero block left over.
Therefore, if we put:
$$H_j = \tilde{R}\,\mathrm{diag}(0,\,0,\,\cdots,\,\Lambda_j,\cdots)\,\tilde{R}^T$$
then $R_j=\exp(H_j)$ with $R_1\circ R_2\circ\,\cdots R_{N\,\mathrm{div}\, 2}$ are then readily seen to make up the decomposition with the properties that John claims, to wit:
- The $R_j$ are each rotations, each which transforms one plane only and each also has a dimension $N-2$ invariant and untransformed space (the analogue of the "axis");
- The planes transformed by the $R_j$ are mutually orthogonal and indeed the planes spanned by the unit vectors $\tilde{R}_j\,\hat{e}_{2\,j}$ and $\tilde{R}_j\,\hat{e}_{2\,j+1}$, where the $\hat{e}_j$ are the orthonormal basis in which all the operators discussed have matrices as written above;
- (as a consequence of 2.) the $R_j$ are mutually commuting.
Thus we can easily see that:
- If the dimension $N$ is odd, there is always a dimension 1 invariant, untransformed space, corresponding to the 1D zero block cited above, further to the invariant spaces described below;
- If the dimension is even, a nontrivial proper orthogonal transformation's untransformed space can be any of the dimensions $0,\,2,\,4,\,\cdots N-2$. The invariant spaces are of dimensions $0,\,2,\,4,\,\cdots,\,N$
This decomposition is about one particular rotation operator and is not to be confused with the notion of Canonical Co-ordinates of the Second Kind (see Chapter 1, Proposition 3.3 of V.V. Gorbatsevich, E.B. Vinberg, "Lie Groups and Lie Algebras I: Foundations of Lie Theory and Lie Transformation Groups", Springer, 2013), which are a generalized notion of Euler Angles. Here, a set of $H_j\in\mathfrak{so}(N)$ for $j=1,\,\cdots,\,N$ (note, there are now $N$ of them, not $N\,\mathrm{div}\,2$ of them) is chosen as a basis, i.e. to span $\mathfrak{so}(N)$. The the following are true:
- The set $\mathbf{G}=\left\{\left.\prod\limits_{j=1}^N\,\exp(\theta_j\,H_j)\,\right|\,\theta_j\in\mathbb{R}\right\}$ contains a neighborhood of the identity in $\mathrm{SO}(N)$;
- If, further, the $H_j$ are orthogonal with respect to the Killing form $\langle X,\,Y\rangle=\mathrm{tr}(\mathrm{ad}(X)\,\mathrm{ad}(Y))$, then the set $\mathbf{G}$ above is the whole of $\mathrm{SO}(N)$.
Property 1, as shown in the Gorbatsevich & Vinberg reference cited above, is a general and fundamental property of all Lie groups (if we replace $\mathfrak{so}(N)$ by the group's Lie algebra and $\mathrm{SO}(N)$ by the group); property 2 holds for compact semisimple ones only.
If the similarity transformation I have here pulled out of thin air seems mysterious, readers may be more familiar with the a re-ordered version of the similarity transformation $\tilde{R}$ above where we decompose a skew-symmetric, closed 2-form $\omega$ in an even dimension case so that its matrix $\Omega$ is:
$$\Omega = \tilde{R}\; \left(\begin{array}{cc}0&-\mathrm{id}_{\frac{N}{2}}\\\mathrm{id}_{\frac{N}{2}}&0\end{array}\right)\;\tilde{R}^T$$
which we implicitly do whenever we label a symplectic space with (in general nonunique) "canonical co-ordinates" so that $\omega$ then has the matrix:
$$\Omega = \left(\begin{array}{cc}0&-\mathrm{id}_{\frac{N}{2}}\\\mathrm{id}_{\frac{N}{2}}&0\end{array}\right)$$
Here we have a different usage of the word "canonical", this time as used in Hamiltonian mechanics. The word "canonical" well and truly needs a well pensioned retirement as it has worked so hard in Physics!
