Classification of linear mappings between indefinite inner product spaces

Let \(\mathcal A:U\to V\) be a linear mapping between vector spaces \(U\) and \(V\) over a field or skew field \(\mathbb F\) with symmetric, or skew-symmetric, or Hermitian forms \(\mathcal B:U\times U\to\mathbb F\) and \(\mathcal C:V\times V\to\mathbb F.\) We classify the triples \((\mathcal A,\mathcal B,\mathcal C)\) if \(\mathbb F\) is \(\mathbb R\), or \(\mathbb C\), or the skew field of quaternions \(\mathbb H\). We also classify the triples \((\mathcal A,\mathcal B,\mathcal C)\) up to classification of symmetric forms and Hermitian forms if the characteristic of \(\mathbb F\) is not 2.