= Solution
The <tangent space> at the identity is the <kernel of a linear map> from part (i):
$$
\boxed{T_IG=\{H:H^TJ+JH=0\}.}
$$
This says precisely that $JH$ is <skew-symmetric>. Since $J^{-1}=J$, multiplication by $J$ identifies this tangent space with the three-dimensional space of skew-symmetric $3\times3$ <matrices>. In entries it is
$$
\boxed{H=\begin{pmatrix}a&b&0\\c&0&-b\\0&-c&-a\end{pmatrix},\qquad a,b,c\in\mathbb R.}
$$
Substitution verifies the equation directly and displays its three independent parameters.
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