What is cryptic? All relations follow from $u_i=0$ for $2i>\dim X$, where $$ u=\operatorname{Sq}^{-1}w=1+w_1+(w_2+w_1^2)+w_1w_2+(w_4+w_1w_3+w_2^2+w_1^4)+\ldots $$ is the total Wu class. (The reason is the fact that $(\operatorname{Sq}^px)[X]=(w_p\smile x)[X]$ for any class $x\in H^{n-p}(X)$, $n=\dim X$.) Explicitly, in small dimensions we have:

$w_1^2+w_2=w_1w_2=0$ for $\dim X=3$,

$w_1w_2=w_1^4+w_2^2+w_1w_3+w_4=0$ for $\dim X=4$.

Of course, these relations generate an ideal in the algebra of Stiefel--Whitney classes invariant under the Steenrod operations. Hence, in dimension $3$ we also have $w_3=w_1^3=0$, i.e., all classes of dimension $3$ are trivial (any $3$-manifold is null cobordant, Rokhlin's theorem). In dimension $4$, we get, in addition, $w_1w_3=w_1^2w_2=0$.

What is cryptic? All relations follow from $u_i=0$ for $2i>\dim X$, where $$ u=\operatorname{Sq}^{-1}w=1+w_1+(w_2+w_1^2)+w_1w_2+(w_4+w_1w_3+w_2^2+w_1^4)+\ldots $$ is the total Wu class. (The reason is the fact that $(\operatorname{Sq}^px)[X]=(u_p\smile x)[X]$ for any class $x\in H^{n-p}(X)$, $n=\dim X$.) Explicitly, in small dimensions we have:

$w_1^2+w_2=w_1w_2=0$ for $\dim X=3$,

$w_1w_2=w_1^4+w_2^2+w_1w_3+w_4=0$ for $\dim X=4$.

Of course, these relations generate an ideal in the algebra of Stiefel--Whitney classes invariant under the Steenrod operations. Hence, in dimension $3$ we also have $w_3=w_1^3=0$, i.e., all classes of dimension $3$ are trivial (any $3$-manifold is null cobordant, Rokhlin's theorem). In dimension $4$, we get, in addition, $w_1w_3=w_1^2w_2=0$.