Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology induced by the classifying map of the $K$-principal bundle $G \to G/K$.
Then $H^\bullet(BK;\mathbb R) = \mathbb R[\alpha_1, \dots, \alpha_r]$ is a polynomial algebra generated by even degree classes $\alpha_1, \dots, \alpha_r$ (where $r = {\rm rk}(K)$) which transgress through the universal bundle to the primitive elements of $H^\bullet(K; \mathbb R)$.
I am interested in the kernel of $w^\bullet$, i.e. the relations that the classes $w^\bullet(\alpha_1), \dots, w^\bullet(\alpha_r)$ satisfy in $H^\bullet(X; \mathbb R)$. There are some obvious relations in high degrees, since $w^n$ vanishes for $n>\dim X$. Also, all classes in degree $\dim X$ are proportional. But what about low degrees? The map $w^2$ is always injective. Is it true that $w^4$ is injective provided $\dim X > 4$? (For some reason I am particularly interested in whether $w^4 $ is injective if $\dim X \geq 8$, but never mind.) What are the lowest degree relations between the $w^\bullet(\alpha_j)$ in general?