Consider an object in $\mathbb{R}^3$. To rotate this object one only needs two directions of control in the following sense. You can get to any orientation by doing a sequence of rotations, each of which is either yaw or roll (as they are known in aviation, see here for details). Notice a control for pitch is not needed because one can roll 90 degrees then yaw the desired amount, then roll back 90 degrees. In this sense only two controls are needed to navigate in 3 dimensions.
This idea can be stated more formally in the following way. Consider the group $G = SO_3(\mathbb{R})$. It can be generated by the subgroups $G_1 = \{R_ø\oplus I_{1×1}| ø \in \mathbb{R}\}$ and $G_2 = \{I_{1×1} \oplus R_ø | ø \in \mathbb{R}\}$ where $R_ø$ denotes rotation by an angle $ø$ in the plane. By generated, I mean to say each element of G can be expressed as a product of a finite list of elements from $G_1$ and $G_2$. Also, $G_1$ and $G_2$ as topologies are locally homeomorphic to $\mathbb{R}$. This last point is important as it corresponds to the notion that each of these subgroups is a single direction of motion that can be controlled.
To put this result in physical terms, this means it would be possible in principle to fly a plane using only two buttons. One that causes the plane to roll clockwise when viewed from inside the cockpit and another that causes it to yaw clockwise when viewed from above (I realize this would not be practical, however that is not the point).
In particular, I am interested in generalizing this idea to $\mathbb{R}^4$, however generalizations to $\mathbb{R}^n$ would be interesting too. Specifically I would like to know how many controls are needed in $\mathbb{R}^4$ in the sense I have defined above.
If there are any key words I can search that will give me information on this topic that would be great.
The key thing here is that if $Y$ and $R$ represent small yaw and roll transformations, respectively, then $YRY^{-1}R^{-1}$ is a transformation that involves a change in pitch. Thus $Y$ and $R$, although they span a two dimensional subspace of $so(3)$, actually manage to generate a 3-dimensional subgroup of $SO(3)$ (which necessarily ends up being all of $SO(3)$).
Now my Lie groups/Lie algebras are a little shaky (the last time I felt I had a solid grip was 1979, when I took my "orals" at Berkeley!), but I'm pretty sure that the answer to your question is "you need three controls". For with two controls (the same Yaw and Roll as before) you can general all matrices of the form $$ \pmatrix{ M & 0 \\ 0 & 1} $$ where where $M \in SO(3)$, and the "0"s indicate lists of three zeroes. Now consider something like Skew -- rotation in the $zw$-plane. With Skew and Yaw, you can, as with Roll and Yaw, compute $SYS^{-1}y^{-1}$, which will be a matrix with nonzero entries in the 4th row and column. With Skew and Roll, you can make matrices with different nonzero entries in the 4th row and column. And with Skew and Pitch (Pitch is available by the previous argument!), you can generate yet another. And that's enough to generate all of $SO(4)$.
Let me put this a little differently: Consider matrices of the form $$ \pmatrix{ 1 & 0 \\ 0 & M} $$ where $M \in SO(3)$. These can be generated by Skew and Something (probably Pitch), just as the others were generated by Yaw and Roll.
And now you just need to see that every matrix in $SO(4)$ is a product of these two special types of matrices, which is pretty clear (I believe).