= Solution
Choose an orthonormal <basis> $e_1,e_2,e_3$ in the real <Euclidean space> containing the <root system>, and take <simple roots>
$$
\alpha_1=e_1-e_2,\qquad\alpha_2=e_2-e_3,\qquad\alpha_3=e_3.
$$
Their squared lengths are $2,2,1$. Use the <Cartan matrix> convention $A_{ij}=2(\alpha_i,\alpha_j)/(\alpha_j,\alpha_j)$, with the denominator indexed by the column. Direct <inner products> give
$$
\boxed{A(B_3)=\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix},\qquad
A(A_3)=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix}.}
$$
The <Dynkin diagram> of the <B3 root system> is a three-node chain with a single bond from node 1 to node 2 and a double bond from node 2 to the short node 3; the arrow points toward node 3. The <A3 root system> has a chain of three nodes joined by single bonds. A convention putting the row length in the denominator transposes the first displayed <Cartan matrix>.
For the <Extended Dynkin diagram>, the <highest root> is $\theta=e_1+e_2=\alpha_1+2\alpha_2+2\alpha_3$. Add $\alpha_0=-\theta$. Its only nonzero off-diagonal <inner product> among these <simple roots> is $(\alpha_0,\alpha_2)=-1$. Deleting the short node 3 leaves the chain $0-2-1$, with three equal-length <roots>, which is $A_3$.
\Image[/past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2018-dynkin-diagrams.png]
{title=Finite Dynkin diagrams and the long-root subalgebra of the extended B3 diagram}
{description=Double-bond arrows point toward the short <root>. The extended <Dynkin diagram> adds $\alpha_0=-\theta$. Deleting node 3 leaves the blue chain $0-2-1$, of type $A_3$.}
More concretely, the long <roots> $\pm e_i\pm e_j$ form the <D3 root system>, which is isomorphic to the <A3 root system>. The corresponding <root spaces> and <Cartan subalgebra> generate
$$
\boxed{\mathfrak{su}(4)\cong\mathfrak{so}(6)\subset\mathfrak{so}(7)}
$$
for the compact real forms. At the matrix level the inclusion fixes one coordinate in seven-dimensional space. This establishes the required <Lie subalgebra> inclusion; the associated global groups require the distinction explained in <4/d>.
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