Let \( \mathfrak{a} \) be an algebraic Lie algebra. An adapted pair for \( \mathfrak{a} \) is a pair (h, η) consisting of an ad-semisimple element of h ∈ \( \mathfrak{a} \) and a regular element of η ∈ \( \mathfrak{a} \) ∗ satisfying (ad h)η = −η. In general such pairs are not easy to find and even more difficult to classify. A natural question is whether ad h has integer eigenvalues on \( \mathfrak{a} \), a property called the integrality of the adapted pair. In general this fails even for a Frobenius subalgebra of \( \mathfrak{s}\mathfrak{l} \)(3) and rather seriously in the sense that any rational number may serve as an eigenvalue. Nevertheless, integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra.Call 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, η), then the eigenvalues of ad h on \( \mathfrak{a} \) lie in (1/m)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, η). Let \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \) be the sub-algebra spanned by the eigensubspaces of ad h with integer eigenvalue. It is shown that the canonical truncation of \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \) is regular. Sufficient knowledge of the relation between the generators for the semi-invariant polynomial functions on \( \mathfrak{a}* \) and on \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}}^{*} \) can then lead to establishing that \( \mathfrak{a} \) = \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \). A particular interesting case is when a is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra \( \mathfrak{g} \). If \( \mathfrak{g} \) is of type Α, then integrality already holds by the paragraph above. If \( \mathfrak{g} \) is of type C and \( \mathfrak{a} \) is a truncated parabolic subalgebra then a rather refined analysis shows that \( \mathfrak{a} \) = \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \). In principle this method can also be applied to biparabolic subalgebras in type C but there are some difficult combinatorial questions involving meanders to be resolved. Outside types A and C further technical complications arise out of an insufficient knowledge of the subalgebra of semi-invariant polynomial functions on the dual of a biparabolic.