  Let $\mathfrak a$ be an algebraic Lie algebra. An adapted pair for $\mathfrak
a$ is pair $(h,\eta)$ consisting of an ad-semisimple element of $h \in
\mathfrak a$ and a regular element of $\eta \in \mathfrak a^*$ satisfying $(ad
\ h)\eta=-\eta$. An adapted pair $(h,\eta)$ is said to satisfy integrality if
$ad \ h$ has integer eigenvalues on $\mathfrak a$. Integrality is shown to hold
for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple
Lie algebra; but may fail in general.
  Call $\mathfrak a$ regular if there are no proper semi-invariant polynomial
functions on $\mathfrak a^*$ and if the subalgebra of invariant functions is
polynomial. In this case there are no known counter-examples to integrality. It
is shown that if $\mathfrak a$ is the canonical truncation of a biparabolic
subalgebra of a simple Lie algebra $\mathfrak g$ which is regular and admits an
adapted pair $(h,\eta)$, then the eigenvalues of $ad \ h$ on $\mathfrak a$ lie
in $\frac{1}{m}\mathbb Z$, where $m$ is a coefficient of a simple root in the
highest root of $\mathfrak g$.
  Let $\mathfrak a$ be a regular Lie algebra admitting an adapted pair
$(h,\eta)$. Let $\mathfrak a_\mathbb Z$ be the subalgebra spanned by the
eigensubspaces of $ad \ h$ with integer eigenvalue. It is shown that the
canonical truncation of $\mathfrak a_\mathbb Z$ is regular. Sufficient
knowledge of the relation between the generators for the invariant polynomial
functions on $\mathfrak a^*$ and on $\mathfrak a^*_\mathbb Z$ can then lead to
establishing the integrality of $(h,\eta)$. This method is used to show the
integrality of an adapted pair for a truncated parabolic subalgebra of a simple
Lie algebra of type $C$.
