If $dim scrH=3$, then $dim otimes^3scrH =27$, $dim wedge^3scrH =1$ and $dim vee^3scrH =10$. In terms of an orthonormal basis of $scrH$, write an element of $(wedge^3scrH )oplus vee^3scrH)^perp$.
Solution. Let $e_1,e_2,e_3$ be an orthonormal basis of $scrH$, then $$ex e_1otimes e_1otimes e_1-e_1otimes e_1otimes e_2in (wedge^3scrH )oplus vee^3scrH)^perp. eex$$