Given a Lie algebra$(\mathfrak{g}, [-,-])$ and a Lie algebra representation$\mathfrak{g} \otimes V \overset{\rho}{\to} V$, and given non-degenerate inner products, hence “metrics”, both on $\mathfrak{g}$ and on $V$, one may ask that all structure is compatible with these metrics. For the Lie algebra this means to have a metric Lie algebra and for the representation this means to have an orthogonal representation. Hence together this is an orthogonal representation of a metric Lie algebra, or metric Lie representation, for short.

Definition

The following table shows the data in a metric Lie representation equivalently