Invariant connections with skew-torsion and $\nabla$-Einstein manifolds

For a compact connected Lie group $G$ we study the class of bi-invariant affine connections whose geodesics through $e\in G$ are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\frak{g}$ coincide with the bi-invariant metric connections. Next we describe the geometry of a naturally reductive space $(M=G/K, g)$ endowed with a family of $G$-invariant connections $\nabla^{\alpha}$ whose torsion is a multiple of the torsion of the canonical connection $\nabla^{c}$. For the spheres ${\rm S}^{6}$ and ${\rm S}^{7}$ we prove that the space of ${\rm G}_2$ (resp. ${\rm Spin}(7)$)-invariant affine or metric connections consists of the family $\nabla^{\alpha}$. Then we examine the "constancy" of the induced Ricci tensor ${\rm Ric}^{\alpha}$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a $\nabla^{\alpha}$-Einstein manifold for any $\alpha\in\mathbb{R}$. We also provide examples of $\nabla^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands.