= Solution
For a differentiable curve $Q(t)\in SO(5)$ with $Q(0)=I_5$, differentiate $Q(t)^TQ(t)=I_5$. Its tangent <matrix> $X=Q'(0)$ is real and satisfies $X^T+X=0$. Conversely the <matrix exponential> of any real <antisymmetric matrix> is orthogonal, and $\det e^{tX}=e^{t\operatorname{tr}X}=1$ because $\operatorname{tr}X=0$. Hence the real <Special orthogonal Lie algebra> is
$$
\boxed{\mathfrak{so}(5)=\{X\in M_5(\mathbb R):X^T=-X\},\qquad [X,Y]=XY-YX.}
$$
Let $E_{ab}$ denote a <matrix unit>. A simple real <basis> is
$$
\boxed{M_{ab}=E_{ab}-E_{ba},\qquad 1\leq a<b\leq5.}
$$
Each independent entry above the diagonal supplies one real parameter, giving $\binom52=10$ parameters. The <commutator> of two <basis> <matrices> is
$$
[M_{ab},M_{cd}]=\delta_{bc}M_{ad}-\delta_{ac}M_{bd}-\delta_{bd}M_{ac}+\delta_{ad}M_{bc},
$$
where $M_{ba}=-M_{ab}$ and $M_{aa}=0$.
A <Cartan subalgebra> is spanned by the commuting <matrices> $M_{12},M_{34}$, which generate independent rotations in two orthogonal planes. A generic $h=aM_{12}+bM_{34}$, with $ab(a^2-b^2)\ne0$, has distinct complex <eigenvalues> $ia,-ia,ib,-ib,0$. A real <antisymmetric matrix> commuting with it preserves those two planes and the remaining line; it is a multiple of the rotation in each plane and is zero on the line. Thus the commuting <Lie subalgebra> has <dimension> two and is maximal toral. Therefore
$$
\boxed{\dim_{\mathbb R}\mathfrak{so}(5)=10,\qquad\operatorname{rank}\mathfrak{so}(5)=2.}
$$
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