!newop Sp
!newop SU
!newop SO
!newop st

\renewcommand{\i}{\underline{i}}
\renewcommand{\j}{\underline{j}}
\renewcommand{\k}{\underline{k}}

Recall that the group of unit quaternions $\Sp(1) ~= \SU_2(\C)$ is a double
cover of $\SO_3(\R)$, and locally isomorphic at the identity. Hence the groups
of infinitesimals $\SO_3^{00}$ and 
\[ \Sp(1)^{00} = \{ \alpha + \beta\i + \gamma\j + \delta\k \;|\;
    \alpha^2 + \beta^2 + \gamma^2 + \delta^2 = 1,
    \st(\alpha-1)=\st(\beta)=\st(\gamma)=\st(\delta)=0 \} \]
are isomorphic as groups.

Let $g = c + s\k (- \Sp(1)^{00}$ with $s>0$.

Let $u = \alpha + \beta\i + \gamma\j + \delta\k (- \Sp(1)^{00}$.

Then we calculate
*\[a*
    ug &= 
    \lambda + (\beta c + \gamma s)\i + (\gamma c - \beta s)\j + \mu\k; \\\
    u^{-1}g^{-1} &=
    \lambda + (-\beta c + \gamma s)\i + (-\gamma c - \beta s)\j - \mu\k
    *a\]*
where
*\[a*
    \lambda &= (\alpha c - \delta s); \\\
    \mu &= (\alpha s + \delta c)
    ,*a\]*

and so (for some $\nu (- \mathfrak{m}$)
*\[a* [u,g] &= u^{-1}g^{-1}ug =
(1-\nu)
+ 2(\lambda\gamma - \mu\beta)s\i
+ 2(-\lambda\beta - \mu\gamma)s\j
+ 2(\beta^2+\gamma^2)cs\k \\\
&\in
(1+\mathfrak{m}) + B\i + B\j + 2(\beta^2+\gamma^2)cs\k
*a\]*

where $B$ is the closed valuative ball around 0 of radius $v(s) +
\min\{v(\beta),v(\gamma)\}$.

Hence if $u_1,u_2 (- \Sp(1)^{00}$,
then the term of least valuation in the resulting expression for the $\k$
component $\eta(u_1,u_2)$ of $[u_1,g][u_2,g]$ is
\[ 2(\beta_1^2+\gamma_1^2+\beta_2^2+\gamma_2^2)cs ,\]
since its valuation is
$v(s) + 2\min\{v(\beta_1),v(\gamma_1),v(\beta_2),v(\gamma_2)\}
<
(v(s) + \min\{v(\beta_1),v(\gamma_1)\}) + 
(v(s) + \min\{v(\beta_2),v(\gamma_2)\})$.

Hence the $\k$ component of $[u_1,g][u_2,g]$ is non-negative.


%Now set $\delta=0$ in the above, yielding
%\[ \Im([u,g]) = 2(\alpha\gamma c - \alpha\beta s)s\i
%+ 2(-\alpha\beta c - \alpha\gamma s)s\j
%+ 2(\beta^2+\gamma^2)cs\k .\]
%
%Then 
%\[ [u_1,g][u_2,g] = 
