= Solution
For a real <inner product space> $V=\mathbb R^5$, define
$$
\Phi(u\wedge v)(w)=\langle v,w\rangle u-\langle u,w\rangle v.
$$
This map takes values in the <Special orthogonal Lie algebra>: its matrices are skew-symmetric. It sends $e_i\wedge e_j$ to $E_{ij}-E_{ji}$, so it is an isomorphism between the two ten-dimensional spaces. For $g\in SO(5)$, invariance of the inner product gives
$$
\Phi(gu\wedge gv)=g\Phi(u\wedge v)g^{-1}.
$$
This proves the <exterior square realization of the orthogonal adjoint representation>, including equivariance, rather than merely matching dimensions.
After complexification, part (a) is therefore the <weight decomposition> of the <Adjoint representation>. Its zero <weight space> is the complex centralizer of $\mathfrak t$. The two zero-weight wedges are nonzero multiples of $e_1\wedge e_2$ and $e_3\wedge e_4$, which map under $\Phi$ to the two infinitesimal rotation generators of $T$. Hence
$$
Z_{\mathfrak{so}(5)}(\mathfrak t)=\mathfrak t.
$$
If a larger torus contained $T$, its Lie algebra would commute with $\mathfrak t$ and therefore lie in $\mathfrak t$. Equality of Lie algebras and connectedness of both tori force equality of the groups. Thus \b[$T$ is a maximal torus].
Let $\varepsilon_j$ denote the real coordinate functional $x_j$. In the real weight convention, with the common factor $i$ removed from the differentiated torus weights, the <root system> is
$$
\boxed{\Phi=\{\pm\varepsilon_1,\pm\varepsilon_2,
\pm(\varepsilon_1+\varepsilon_2),\pm(\varepsilon_1-\varepsilon_2)\}.}
$$
These are the <roots of a root system> of type <B2 root system>. In the imaginary-valued compact Lie-algebra convention each displayed functional is multiplied by $i$. Every root space is one-dimensional, as part (a) explicitly shows.
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