The intermediate Lie algebra $\mathfrak d_{n-1/2}$, the weight scheme and finite-dimensional representations with highest weight

Multiple points of the spectrum in the reduction $D_n\downarrow D_{n-1}$ are separated by introducing a non-semisimple intermediate subalgebra and a weight scheme different from the Gel'fand–Tsetlin scheme. We suggest a method of constructing a weight basis in the space of a finite-dimensional irreducible representation of $D_n$. The elements of this basis are labelled by such weight schemes. We also study the category of finite-dimensional highest-weight representations of this intermediate Lie algebra.