spin geometry, string geometry, fivebrane geometry …
In physics, the matrix representing a bilinear form on a real spin representation, i.e. on Majorana spinors is also called the charge conjugation matrix on spinors.
For details see at Majorana spinor – Charge conjugation matrix.
The abstract relation to bilinear forms on spinors is commented on for instance around (3.4) of
and in appendix B.1 of
The matrix component yoga used in physics is summarized for instance in
Antoine Van Proeyen, section 3.1 of Tools for supersymmetry (arXiv:hep-th/9910030)
Joseph Polchinski, volume II, appendix B of String theory
Friedemann Brandt, section 2.1, 2.3 of Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, (arXiv:0911.2118)