if vectors $u,v,w$ are linearly independent, then can we write $u\times(v\times w)=av+bw$ with some scalars $a,b$?
i think that such scalars do exist, because $v×w$ is perpendicular to the plane that is spanned by $v$ and $w$ and thus $u\times(v\times w)$ must lie in that plane
also as all three vectors are linearly independent then their cross products cannot be equal to the zero vector
are my arguments correct? or am i missing something?