Let $L$ be a lie algebra and $H$ an abelian subalgebra of $L$ such that each element of $h \in H$ is diagonalizable under the adjoint representation. So there exists a basis of common eigenvectors for $ad(h)$ for all $h \in H$ and non zero functionals $\lambda_1 \ldots \lambda_n \in H^*$ \begin{gather} L=C_L(H) \oplus \bigoplus_{i=1}^n L_{\lambda_i} \end{gather} With $C_L(H)$ is the centralizer of $H$ and $L_\lambda =\{x \in L: [h,x]=\lambda(h)x, \forall h\in H\} $. Now if $H$ is maximal among the abelian lie sub-algebras that act diagonalizable on $L$ via the adjoint representation, the functionals $\lambda_i$ are a root system for $H^*$ and $H$ is self-centralizing. What can we say if $H$ is not maximal?
For examble, if we don´t know that $H$ is maximal but we find out that the $\lambda_i$ are a root system for $H^*$, can we say that the decomposition of $L$ above is a root decomposition and $H$ self-centralizing? In general, are there some ways to obtain information about how $H$ is embedded in $L$ from the eigenvalues $\lambda_i$?