= Solution
Preserving the symmetric <bilinear form> infinitesimally means $Z^TM+MZ=0$. In blocks of sizes $2,2,1$, write $Z$ as a general block matrix and multiply by the given $M$. The condition says that $MZ$ is skew-symmetric. It forces the lower-right entry to be zero, the middle diagonal block to be the negative transpose of the upper-left block, and the two off-diagonal two-by-two blocks to be skew-symmetric. Thus the general element of the <Special orthogonal Lie algebra> is
$$
\boxed{Z=\begin{pmatrix}
A&B&p\\
C&-A^T&q\\
-q^T&-p^T&0
\end{pmatrix},\qquad B^T=-B,\ C^T=-C,}
$$
where $A$ is an arbitrary two-by-two <matrix> and $p,q$ are arbitrary two-component columns. More explicitly, using ten independent complex parameters,
$$
Z=\begin{pmatrix}
a&b&0&e&p_1\\
c&d&-e&0&p_2\\
0&f&-a&-c&q_1\\
-f&0&-b&-d&q_2\\
-q_1&-q_2&-p_1&-p_2&0
\end{pmatrix}.
$$
Conversely, substituting this matrix into $Z^TM+MZ$ gives zero, so there are no further restrictions. In particular $\dim\mathfrak{so}_5=4+1+1+2+2=10$.
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