= Solution
The <complexification of a Lie algebra> gives $\mathfrak{su}(4)_{\mathbb C}=\mathfrak{sl}_4(\mathbb C)$, of type <A3 root system>. Choose <simple roots> $\alpha_1=e_1-e_2$, $\alpha_2=e_2-e_3$, $\alpha_3=e_3-e_4$, with squared lengths two. The <Cartan matrix> and <Dynkin diagram> are
$$
\boxed{A=\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix},\qquad
\underset{1}{\circ}\! -\! \underset{2}{\circ}\! -\! \underset{3}{\circ}.}
$$
Each line represents a single bond. In the <Dynkin label> convention $w_a=\langle w,\alpha_a^\vee\rangle$, the <simple roots> have coordinates equal to the rows of $A$.
In the defining <representation>, apply the <simple coroots> $E_{aa}-E_{a+1,a+1}$ to the four coordinate <basis vectors>. Their <weights> are
$$
\boxed{\mathbf4=(1,0,0):\quad
v_1=(1,0,0),\quad v_2=(-1,1,0),\quad v_3=(0,-1,1),\quad v_4=(0,0,-1).}
$$
Equivalently these are obtained by subtracting $\alpha_1,\alpha_2,\alpha_3$ successively from the <highest weight>. The <dual representation> has the negative <weights>, hence
$$
\boxed{\overline{\mathbf4}=(0,0,1):\quad
(-1,0,0),\quad(1,-1,0),\quad(0,1,-1),\quad(0,0,1).}
$$
All listed <weight multiplicities> are one, so both <dimensions> are four.
The <highest weight> $(1,0,1)$ is $\alpha_1+\alpha_2+\alpha_3$, the <highest root>. Its <irreducible representation> is therefore the <Adjoint representation>. Under the action on <traceless matrices> by conjugation, the off-diagonal <matrix unit> $E_{ij}$ has <weight> $v_i-v_j$, while the diagonal <Cartan subalgebra> has zero <weight>. Thus
$$
\boxed{\begin{aligned}
\mathbf{15}=(1,0,1):\quad
&\pm(2,-1,0),\quad\pm(-1,2,-1),\quad\pm(0,-1,2),\\
&\pm(1,1,-1),\quad\pm(-1,1,1),\quad\pm(1,0,1),\\
&(0,0,0)^{[3]}.
\end{aligned}}
$$
The superscript $[3]$ indicates <weight multiplicity> three. The twelve nonzero <weights> each have <weight multiplicity> one, giving $12+3=15$ in total. This also follows from the <tensor product representation> $\mathbf4\otimes\overline{\mathbf4}=\mathbf{15}\oplus\mathbf1$: the sixteen differences $v_i-v_j$ contain four zero <weights>, and removing the identity <matrix> removes one of them.
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