Indecomposable $1$-factorizations of the complete multigraph $\lambda K_{2n}$ for every $\lambda\leq 2n$

A $1$-factorization of the complete multigraph $\lambda K_{2n}$ is said to be indecomposable if it cannot be represented as the union of $1$-factorizations of $\lambda_0 K_{2n}$ and $(\lambda-\lambda_0) K_{2n}$, where $\lambda_0<\lambda$. It is said to be simple if no $1$-factor is repeated. For every $n\geq 9$ and for every $(n-2)/3\leq\lambda\leq 2n$, we construct an indecomposable $1$-factorization of $\lambda K_{2n}$ which is not simple. These $1$-factorizations provide simple and indecomposable $1$-factorizations of $\lambda K_{2s}$ for every $s\geq 18$ and $2\leq\lambda\leq 2\lfloor s/2\rfloor-1$. We also give a generalization of a result by Colbourn et al. which provides a simple and indecomposable $1$-factorization of $\lambda K_{2n}$, where $2n=p^m+1$, $\lambda=(p^m-1)/2$, $p$ prime.