Simple examples of $3 \times 3$ rotation matrices I'd like to have some numerically simple examples of $3 \times 3$ rotation matrices that are easy to handle in hand calculations (using only your brain and a pencil). Matrices that contain too many zeros and ones are boring, and ones with square roots are undesirable. A good example is something like
$$
M = \frac19 \begin{bmatrix}
1 & -4 & 8 \\
8 &  4 & 1  \\
-4 & 7 & 4
\end{bmatrix}
$$
Does anyone have any other examples, or a process for generating them?
One general formula for a rotation matrix is given here. So one possible approach would be to choose $u_x$, $u_y$, $u_z$ and $\theta$ so that you get something simple. Simple enough for hand calculations, but not trivial. Like the example given above.

Here are some:
$$\left[ \begin {array}{ccc} 1/3&2/3&2/3\\ 2/3&-2/3&1/3\\ 2/3&1/3&-2/3\end {array} \right] 
$$
$$\left[ \begin {array}{ccc} 2/7&3/7&6/7\\ 3/7&-6/7&2/7\\ 6/7&2/7&-3/7\end {array} \right] 
$$
$$
\left[ \begin {array}{ccc} \frac{2}{11}&{\frac {6}{11}}&{\frac {9}{11}}\\ -{\frac {6}{11}}&-{\frac {7}{11}}&{\frac {6}{11}}
\\ {\frac {9}{11}}&-{\frac {6}{11}}&\frac{2}{11}\end {array}
 \right] 
$$
EDIT: here are the small positive integer solutions of $a^2 + b^2 + c^2 = d^2$ with
$a \le b \le c$ and $\gcd(a,b,c,d)=1$, in order of increasing $b^2 + c^2$:
$$\eqalign{1^2 + 2^2 + 2^2 &= 3^2\cr
2^2 + 3^2 + 6^2 &= 7^2\cr
4^2 + 4^2 + 7^2 &= 9^2\cr
1^2 + 4^2 + 8^2 &= 9^2\cr
6^2 + 6^2 + 7^2 &= 11^2\cr
2^2 + 6^2 + 9^2 &= 11^2\cr
3^2 + 4^2 + 12^2 &= 13^2\cr
2^2 + 10^2 + 11^2 &= 15^2\cr
2^2 + 5^2 + 14^2 &= 15^2\cr
8^2 + 9^2 + 12^2 &= 17^2\cr
1^2 + 12^2 + 12^2 &= 17^2\cr
}$$
