- J^2 J_0)\inv (\bfone +JJ_0) &=&
\cr
&&
(J_0 + J)\inv (\bfone +JJ_0) &=& 
\bigl( (\bfone -JJ_0) J_0 \bigr)\inv (\bfone +JJ_0) &=& 
\cr
&&
J_0\inv(\bfone -JJ_0)\inv (\bfone +JJ_0) &=& 
- J_0(\bfone -JJ_0)\inv (\bfone +JJ_0) &=& - J_0 W
\endmatrix$$

[*Step 4. $J \in \jj_\omega$ implies $\bfone - W^\sft W \gg 0$ for $W= 
\sfL(J)$*]{}, the operator $\bfone - W^\sft W$ is positively definite, $g(v,v) > g(Wv,Wv)$ for any non-zero $v \in V$. The conjugation of $W \in 
\endo(V)$ is done using the metric $g$.

First we note that $J_0^\sft = J_0\inv =-J_0$. Since $\bfone -JJ_0$ is invertible, the positivity of $\bfone - W^\sft W$ is equivalent to the positivity of $(\bfone -JJ_0)^\sft(\bfone - W^\sft W) (\bfone -JJ_0)$. Computing this operator we obtain $$\matrix\format \c \vbox to 11.5pt{} \\
(\bfone -JJ_0)^\sft(\bfone - W^\sft W) (\bfone -JJ_0) = 
(\bfone -JJ_0)^\sft(\bfone -JJ_0) - 
\cr
(\bfone -JJ_0)^\sft\bigl( (\bfone +JJ_0) (\bfone -JJ_0)\inv \bigr)^\sft
\bigl( (\bfone +JJ_0) (\bfone -JJ_0)\inv \bigr)(\bfone -JJ_0)=
\cr
(\bfone -JJ_0)^\sft(\bfone -JJ_0) -
(\bfone +JJ_0)^\sft (\bfone +JJ_0) =
\cr
(\bfone +J_0 J^\sft)(\bfone - JJ_0) - (\bfone - J_0 J^\sft)(\bfone +JJ_0)=
\cr
\bfone +J_0 J^\sft - JJ_0  -  J_0 J^\sft JJ_0 -
\bfone + J_0 J^\sft - JJ_0 +  J_0 J^\sft JJ_0 =
\cr
