Some four-dimensional orthogonal invariants

Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$.

Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$.

We compute the invariant ring $\mathbb{F}_p[V\oplus V^*]^G$ via explicitly exhibiting a minimal generating set.

Our method finds an application of $s$-invariants appeared in covariant theory of finite groups.