= Solution
Let $N$ be the subgroup whose upper-left block is $I_2$. It is a finitely generated nilpotent subgroup of the integer <upper unitriangular group> $UT_4(\mathbb Z)$ and is normal in $L$. The block-diagonal matrix
$$
t=\operatorname{diag}(A,I_2)
$$
generates an infinite cyclic quotient, so
$$
L=N\rtimes\langle t\rangle.
$$
Finitely generated nilpotent groups are polycyclic, and an extension of polycyclic groups is polycyclic. Hence $L$ is polycyclic.
Inside $L$, retain only $t$ and the entries in positions $(1,3)$ and $(2,3)$. They form a subgroup
$$
\mathbb Z^2\rtimes_A\mathbb Z.
$$
The characteristic polynomial of $A$ is $\lambda^2-5\lambda+1$, so its eigenvalues are
$$
\frac{5\pm\sqrt{21}}2.
$$
One has modulus greater than one, and the resulting semidirect product has <exponential growth of a group>[exponential growth]. Every finitely generated virtually nilpotent group has polynomial growth, as does each of its finitely generated subgroups. Therefore $L$ cannot be virtually nilpotent.
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