information needed to construct the level sets of Casimir invariants is contained in the differential one-forms that are annuled by the vectors tangent to these sets. Obviously, if we explicitly knew a Casimir invariant $C$, $\langle {\rm d}C,{\rm J}{\rm d}g \rangle = 0$ tells us that d$C$ is such a one-form. However, we do not need to know the Casimir invariants to find differential one-forms that are annulled by these vectors. In fact, from $\langle {\rm d}g,{\rm J}{\rm d}C \rangle = \langle 
{\rm d}C,{\rm J}{\rm d}g \rangle = 0,$ it is clear that any element in the kernel of J will do. And so, we can get the information we need to determine all of the Casimir invariants by finding all the linearly independent null covectors of J.

The null covectors of J, call them $\gamma ^{(i)}$ (suppose there are $n$), are differential one-forms. They are not necessarily exact: that is, it is not necessarily true that there exists an $F$ such that $\gamma ^{(i)} = 
{\rm d}F.$

But once we have the $\gamma ^{(i)}$, we can find the Casimir invariants from the condition that the $\gamma ^{(i)}$ vanish when paired with any vector tangent to a level set of 