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Let $F$ be a field of characteristic different from two. In the convention corresponding to isometric structures with no eigenvalues $\pm 1$, use finite even-sized matrices $V$ for which both $V-V^T$ and $V+V^T$ are nonsingular. Their Witt group under direct sum, with metabolic Seifert forms set to zero, is the algebraic concordance group over $F$. Thus $V$ and $W$ define the same class when $V\oplus (-W)$ is metabolic.^[ref]

Invertibility of $V$ itself is a separate condition: in this Witt classification one can choose invertible representatives before converting to isometric structures. For $F=\mathbb{Q}$, integral Seifert matrices of knots satisfy the symmetric and skew nondegeneracy conditions, and define rational algebraic-concordance classes. Extension to $\mathbb{R}$ detects signature information, while extension to p-adic fields can retain torsion subgroup information invisible over $\mathbb{R}$.

A metabolizer for an integral Seifert form $\theta$ on a free abelian group $L$ of rank $2g$ is a rank-$g$ direct summand $M\subseteq L$ on which the form vanishes:This is the data witnessing a metabolic Seifert form. Over a field, the corresponding definition uses a half-dimensional vector subspace. The same term also denotes a half-dimensional isotropic subspace for a metabolic bilinear form.

For $L={\mathbb{Z}}^2$ with Seifert matrix $V=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, the subgroup spanned by the first basis vector is a metabolizer. Its self-pairing is zero and it has the complementary summand spanned by the second basis vector. The subgroup generated by twice the first basis vector is also isotropic and has rank one, but is not a direct summand: its quotient in $L$ has torsion.

Thus half rank alone does not suffice over the integers. The direct-summand condition ensures that a basis of the metabolizer extends to a basis of the whole lattice, exposing a zero $g$-by-$g$ block in the Seifert matrix.