The maximal rotation matrix Let's consider two numbers calculated for a rotation matrix which are:      


*

*$s_e=$   the sum of all entries of a matrix  

*$s_a=$   the sum of absolute values of all entries for a given matrix.
It would be interesting to know for what rotation matrix these numbers are maximal.
Case for 2D is rather straightforward, it is known general form of a matrix and both numbers $s_e$,  $s_a$ are just  functions of a single variable  $\theta$
$ 
        R= \begin{bmatrix}
           \cos(\theta) & -\sin(\theta)   \\  
          \ sin(\theta)  & \cos(\theta)  \\ 
       \end{bmatrix}
$    
Case for 3D seems much more complicated.  We have 3 variables ($3$ DOF) and final form is quite complicated. Additionally it's hard to guess what form of equation we should take:  Rodrigues formula?  Euler rotations? How  to use constraints (highly symmetrical in their structure) usually imposed on a rotation matrix ?
Examples shed some   light on the problem.   
Let's look at the identity matrix. Here situation is simple.
   $$R=\begin{bmatrix}
   1 &     0   &    0   \\
   0 &    1    &   0    \\
   0 &   0   &    1  \\
      \end{bmatrix}  $$ 
 $s_e= 3, s_a=3 $   
Other example 
  $$R=\begin{bmatrix}
    \dfrac{\sqrt{2}}{2} &   -\dfrac{\sqrt{2}}{2}     &    0   \\
   \dfrac{\sqrt{2}}{2} &   \dfrac{\sqrt{2}}{2}   &  0   \\
   0 &    0  &    1   \\
      \end{bmatrix}  $$   
We have here  
$s_e= 1+\sqrt{2}, s_a=1+2\sqrt{2}$.    
The number  $s_e$ is less  and  $s_a$ greater comparing to the  case for identity matrix.
Intuition tells that such  a matrix , I call it here "the maximal" rotation matrix and denote appropriately $R_{max(s_e)}$ and $R_{max(s_a)}$, should have entries somehow evenly located in rows and columns and negative entries should be as small as possible. $0$'s  should be rather absent.
Good candidate for it in the second sense, it seems, is the matrix mentioned earlier by me (A certain unique rotation matrix).
$$A=\begin{bmatrix}
   -\dfrac{1}{3} &     \dfrac{2}{3}      &     \dfrac{2}{3}   \\
    \dfrac{2}{3}  &    -\dfrac{1}{3}   &   \dfrac{2}{3}    \\
   \dfrac{2}{3} &    \dfrac{2}{3}  &    -\dfrac{1}{3}  \\
      \end{bmatrix}  $$
 For it $s_e= 3, s_a=5 $


*

*Maybe it is the maximal matrix taking into account $s_e$ but how to prove it ? 

*If not what is the maximal 3D rotation   matrix in both senses mentioned above ?

*Could we at least prove that $R_{max(s_a)}$ is from the family of rotations $R_{max(s_e)}$ ?

($App.^*$One can consider also a minimal versions of rotation matrices: $R_{min(s_e)}$ and $R_{min(s_a)}$.  For them values are probably  $-3$  and $3$).
 A: $\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\langle #1\rangle}$Let $A$ be an $n \times n$ orthogonal matrix, and put $\Basis = (1, 1, \dots, 1)$. The sum of the entries of $A$ is the inner product
$$
\Brak{\Basis, A\Basis} = n\cos\theta,
$$
with $\theta$ the angle between $\Basis$ and $A\Basis$. Since there exists an orthogonal matrix fixing $\Basis$ (i.e., with $\theta = 0$), it follows that the maximum sum-of-entries for an $n \times n$ orthogonal matrix is $n$, and this is achieved precisely by the copy of $O(n-1)$ that fixes $\Basis$.

Offhand I don't see any nice way to get at the maximum absolute value sum in general. For $n = 2$, the maximum absolute value sum is clearly $2\sqrt{2}$, so there exist (block diagonal) matrices in $O(2n)$ with absolute value sum $2n\sqrt{2}$, and matrices in $O(2n+1)$ with absolute value sum at least $1 + 2n\sqrt{2}$.
These lower bounds on the maximum absolute value sum are not optimal for $n \geq 4$: For example, the $4 \times 4$ orthogonal matrix
$$
\tfrac{1}{2}\left[\begin{array}{@{}rrrr@{}}
1 & -1 & -1 & -1 \\
1 & -1 &  1 &  1 \\
1 &  1 & -1 &  1 \\
1 &  1 &  1 & -1 \\
\end{array}\right]
$$
has absolute value sum $8 > 4\sqrt{2}$.
For the case $n = 3$, it's plausible your matrix maximizes the absolute value sum: Naively, one expects the set of columns of an extremal matrix to be invariant under rotation about a diagonal axis of the unit cube. It's easy to check your matrix has this symmetry, and its columns are equidistant from the coordinate planes.
