Group Generated by 45-Degree Rotations? What is the group generated by 45-degree rotations about the (I, J, K) axes in three-dimensional space? Is it a finite group? 

Interesting question. The generators are
$$A=\pmatrix{1/\sqrt2&1/\sqrt2&0\\-1/\sqrt2&1/\sqrt2&0\\0&0&1},$$
$$B=\pmatrix{1/\sqrt2&0&1/\sqrt2\\0&1&0\\-1/\sqrt2&0&1/\sqrt2}$$
and
$$C=\pmatrix{1&0&0\\0&1/\sqrt2&1/\sqrt2\\0&-1/\sqrt2&1/\sqrt2}.$$
Then
$$AB=\pmatrix{1/2&1/\sqrt2&1/2\\-1/2&1\sqrt2&-1/2\\-1\sqrt2&0&1/\sqrt2}.$$
$AB$ has trace $1/2+\sqrt2$. This is not an algebraic integer.
The trace of any orthogonal matrix of finite order is the sum of
roots of unity and is an algebraic integer. So $AB$ has infinite order
and therefore $A$ and $B$ generate an infinite group.
