= Solution
For the <A5 root system>, take <simple roots> $\alpha_i=e_i-e_{i+1}$, $i=1,\ldots,5$, in the hyperplane $\sum_{i=1}^6x_i=0$. Their squared lengths are two, adjacent <simple roots> have <inner product> $-1$, and all other pairs are orthogonal. Hence the <An Dynkin diagram> is the five-node single-edge chain
$$
\underset{1}{\circ}\! -\!\underset{2}{\circ}\! -\!\underset{3}{\circ}\! -\!\underset{4}{\circ}\! -\!\underset{5}{\circ}.
$$
Its <Cartan matrix> is
$$
\boxed{A(A_5)=\begin{pmatrix}
2&-1&0&0&0\\
-1&2&-1&0&0\\
0&-1&2&-1&0\\
0&0&-1&2&-1\\
0&0&0&-1&2
\end{pmatrix}.}
$$
The complex <Lie algebra> is $\mathfrak{sl}_6(\mathbb C)$; the corresponding compact <special unitary group> is $SU(6)$, whose <Lie algebra> is its compact real form.
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