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 $V^{\ast }$. We compute the invariant ring ${\mathbb{F}}_p[V\oplus V^{\ast }{]}^G$ via explicitly exhibiting a minimal generating set. Our method finds an application of $s$-invariants appeared in covariant theory of finite groups.