Let $T\in\mathcal{L}({\mathbb{C}^4})$ be the linear operator that we wish to determine (i.e., $T: \mathbb{C}^4 \to \mathbb{C}^4$). As $T$ is defined in a complex vector space, $\mathbb{C}^4$ will admit a decomposition in terms of the null spaces defining the general eigenvectors of $T$.
$$ \mathbb{C}^4 = \text{null}(T-\lambda_1)^{d_1} \hspace{5pt}\oplus \hspace{5pt}... \hspace{5pt}\oplus \hspace{5pt} \text{null}(T-\lambda_k)^{d_k} $$
where $d_1,...,d_k$ count the multiplicities of each eigenvalue and $d_1 + ... + d_k = 4$. The characteristic polynomial $p_c(z) : \mathbb{C} \to \mathbb{C}$ is then defined as $$p_c(z) = (z-\lambda_1)^{d_1} ... (z-\lambda_k)^{d_k}$$
We can define the minimal polynomial as follows. The space $\mathcal{L} (\mathbb{C}^4)$ is parameterized by matrices possessing a one in a single location and zeros elsewhere. For example, one such basis vector will be the matrix
$$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} $$
Consequently, $\text{dim}(\mathcal{L}(\mathbb{C}^4)) = \text{dim}(\mathbb{C}^4)^2 = 16$. The list of operators given by $$ (I, T, T^2, ..., T^{n^2}) $$ will posses $n+1$ elements and cannot be linear independent. Selecting $m$ to be the smallest such index that the list is linearly independent, we find that $\exists a_0, ..., a_{m-1} \in \mathbb{C}$ s.t.
$$ a_0 I + a_1 T + ... + a_{m-1} T^{m-1} + T^{m} = 0 $$
which shall yield for us a minimal polynomial $p_m (z) : \mathbb{C} \to \mathbb{C}$ of the form
$$ p_m(z) = a_0 + a_1 z + ... + a_{m-1} z^{m-1} + z^m $$
(In fact the Cayley-Hamilton theorem improves this bound by restricting the total number of terms to $\text{dim}(V) = 4$.
By the fundamental theorem of algebra, we may factor $$ p_c(z) = (z - \lambda_1)^{d_1} ... (z-\lambda_k)^{d_k} $$ $$ p_m(z) = (z- \lambda_1') ... (z-\lambda_4') $$
As every root of the minimal polynomial $p_m(z)$ is inherently an eigenvalue every $\lambda'$ may be identified with some lambda neglecting multiplicity.
Question: Given $p_c(z)$ and $p_m(z)$ in factored form, how may we determine a linear operator $T$ on $\mathbb{C}^4$ whose characteristic polynomial is $p_c(z)$ and whose minimal polynomial is $p_m(z)$?
Remark: I have noticed (from dumb luck) that a matrix of the form $$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & c_1 \\ 0 & 1 & 0 & c_2 \\ 0 & 0 & 1 & c_3 \end{pmatrix} $$ for undetermined coefficients $c_1, ..., c_3$ will work for minimal polynomials of degree less than three.