  We show new algorithms and constructions over linear delta-matroids. We
observe an alternative representation for linear delta-matroids, as a
contraction representation over a skew-symmetric matrix. This is equivalent to
the more standard "twist representation" up to $O(n^\omega)$-time
transformations, but is much more convenient for algorithmic tasks. For
instance, the problem of finding a max-weight feasible set now reduces directly
to the problem of finding a max-weight basis in a linear matroid. Supported by
this representation, we provide new algorithms and constructions over linear
delta-matroids. We show that the union and delta-sum of linear delta-matroids
define linear delta-matroids, and a representation for the resulting
delta-matroid can be constructed in randomized time $O(n^\omega)$. Previously,
it was only known that these operations define delta-matroids. We also note
that every projected linear delta-matroid can be represented as an elementary
projection. This implies that several optimization problems over (projected)
linear delta-matroids, including the coverage, delta-coverage, and parity
problems, reduce (in their decision versions) to a single $O(n^{\omega})$-time
matrix rank computation. Using the methods of Harvey, previously used by
Cheung, Lao and Leung for linear matroid parity, we furthermore show how to
solve the search versions in the same time. This improves on the $O(n^4)$-time
augmenting path algorithm of Geelen, Iwata and Murota. Finally, we consider the
maximum-cardinality delta-matroid intersection problem. Using Storjohann's
algorithms for symbolic determinants, we show that such a solution can be found
in $O(n^{\omega+1})$ time. This is the first polynomial-time algorithm for the
problem, solving an open question of Kakimura and Takamatsu.
