Full-index and ten-component conventions

Let

All spacetime raising and lowering use $\eta$. Distributional pairings are complex bilinear, with flat coordinate densities. Complex polarization is bilinear, not sesquilinear.

Let $S={\mathrm{Sym}}^2(({\mathbb{R}}^4{)}^{\ast })$. For $I=(i,j)$, $0\le i\le j\le 3$, define

Thus

An off-diagonal basis tensor has two entries equal to one; its algebraic dual has two entries equal to $1/2$. The algebraic dual is not simply the raised basis tensor.

Trace reversal is

Put

The subscripts in ${\mathsf{A}}^{\mu \nu }{}_L$ and ${\mathsf{b}}_L$ mean evaluation on $e_L$; they do not prescribe an extra fiber-metric contraction.