= Solution
Equip the finite-dimensional real vector space $V$ with a positive-definite <inner product>. A reduced crystallographic <root system> is a finite set $R\subset V\setminus\{0\}$ spanning $V$ such that, for every $\alpha\in R$,
$$
R\cap\mathbb R\alpha=\{\alpha,-\alpha\},\qquad
s_\alpha(R)=R,\qquad
\frac{2(\beta,\alpha)}{(\alpha,\alpha)}\in\mathbb Z\quad(\beta\in R),
$$
where $s_\alpha(v)=v-2(v,\alpha)\alpha/(\alpha,\alpha)$ is the <orthogonal reflection> in $\alpha^\perp$. The <Weyl group> is $W(R)=\langle s_\alpha:\alpha\in R\rangle$. It permutes the finite set $R$. Its action is faithful because $R$ spans $V$: an element fixing every <root> fixes every vector. Hence it embeds in the finite <symmetric group> of $R$ and \b[is finite].
A <Cartan subalgebra> is a nilpotent self-normalizing <Lie subalgebra>. For a finite-dimensional complex <semisimple Lie algebra> it is equivalently a maximal toral subalgebra: a maximal abelian subalgebra whose elements have <diagonalizable> adjoint actions. The commuting semisimple operators $\operatorname{ad}(h)$ give the <root-space decomposition>
$$
\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in R}\mathfrak g_\alpha,\qquad
\mathfrak g_\alpha=\{x:[h,x]=\alpha(h)x\text{ for every }h\in\mathfrak h\}.
$$
The nonzero weights $\alpha$ are the roots. Their real span inside the complex dual $\mathfrak h^*$, with the positive-definite real <inner product> induced by the <Killing form>, is the Euclidean space in which they form a reduced crystallographic <root system>. In particular one uses this real span, rather than treating all of $\mathfrak h^*$ as a real vector space of twice the <rank>.
For the specified <Special orthogonal Lie algebra>, put $\bar i=7-i$. The <matrix> relation is $x_{ij}=-x_{\bar j,\bar i}$. Thus its diagonal <Cartan subalgebra> consists of
$$
H(t)=\operatorname{diag}(t_1,t_2,t_3,-t_3,-t_2,-t_1).
$$
Let $\varepsilon_i(H(t))=t_i$. The <matrix> units satisfy $[H,E_{ij}]=(d_i-d_j)E_{ij}$, where $d=(t_1,t_2,t_3,-t_3,-t_2,-t_1)$. A <basis> of <root> vectors is supplied, for $1\le i<j\le3$, by
$$
\begin{array}{c|c}
\text{root}&\text{root vector}\\\hline
\varepsilon_i-\varepsilon_j&E_{ij}-E_{\bar j,\bar i}\\
-\varepsilon_i+\varepsilon_j&E_{ji}-E_{\bar i,\bar j}\\
\varepsilon_i+\varepsilon_j&E_{i,\bar j}-E_{j,\bar i}\\
-\varepsilon_i-\varepsilon_j&E_{\bar j,i}-E_{\bar i,j}
\end{array}
$$
These are twelve independent off-diagonal matrices. Together with the three diagonal generators, they account for the full <dimension> $6\cdot5/2=15$, so no further roots occur. Consequently
$$
\boxed{R=\{\pm\varepsilon_i\pm\varepsilon_j:1\le i<j\le3\},\qquad R\text{ has type }D_3.}
$$
Choose the simple roots
$$
\alpha_1=\varepsilon_1-\varepsilon_2,\qquad
\alpha_2=\varepsilon_2-\varepsilon_3,\qquad
\alpha_3=\varepsilon_2+\varepsilon_3.
$$
The six positive roots are $\alpha_1,\alpha_2,\alpha_3,\alpha_1+\alpha_2,\alpha_1+\alpha_3,\alpha_1+\alpha_2+\alpha_3$, and every other <root> is their negative. In the standard orthonormal coordinates all roots have squared length $2$, and their <Cartan matrix> is
$$
\begin{pmatrix}2&-1&-1\\-1&2&0\\-1&0&2\end{pmatrix}.
$$
Its <Dynkin diagram> is the connected chain $\alpha_2-\alpha_1-\alpha_3$. A decomposition into mutually orthogonal nonempty <root> systems would separate this connected set of simple roots, which is impossible. Thus the <root system> is irreducible. The chain is also the diagram $A_3$, which is the <root system> of $\mathfrak{sl}_4(\mathbb C)$. The classification of complex semisimple Lie algebras by their <root> systems yields the <Lie algebra isomorphism>
$$
\boxed{\mathfrak{so}_6(\mathbb C)\cong\mathfrak{sl}_4(\mathbb C).}
$$
An explicit explanation of this coincidence uses the <exterior-power Lie algebra representation> of $\mathfrak{sl}_4$ on $\bigwedge^2\mathbb C^4$, a six-dimensional space. After choosing a volume form, $u\wedge v$ defines a nondegenerate <symmetric bilinear form> on that <exterior square>. The induced action preserves it because a traceless <endomorphism> acts trivially on the top exterior power. This gives a nonzero homomorphism $\mathfrak{sl}_4\to\mathfrak{so}_6$. Its <kernel> is an ideal in the simple algebra $\mathfrak{sl}_4$, so it is zero; equal dimensions $15$ then make it an isomorphism. Over $\mathbb C$ all nondegenerate symmetric forms in <dimension> six are equivalent, including the form with <matrix> $J$.
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