η-symbols in exceptional field theory

We present the universal form of $\eta$-symbols that can be applied to an arbitrary $E_{d\left(d\right)}$ exceptional field theory (EFT) up to $d=7$. We then express the $Y$-tensor, which governs the gauge algebra of EFT, as a quadratic form of the $\eta$-symbols. The usual definition of the $Y$-tensor strongly depends on the dimension of the compactification torus while it is not the case for our $Y$-tensor. Furthermore, using the $\eta$-symbols, we propose a universal form of the linear section equation. In particular, in the $SL\left(5\right)$ EFT, we explicitly show the equivalence to the known linear section equation.