Invariant theory of the indefinite orthogonal groups

Question

I believe the following statements are true:

Let $V$ be a finite-dimensional real vector space with a positive-definite inner product $g$. Let $g_{\otimes n}$ denote the natural extension of $g$ to $V^{\otimes n}=V\otimes\cdots\otimes V.$ Let $\rho:\text{O}(V,g)\to\text{O}(V^{\otimes n},g_{\otimes n})$ be the standard group homomorphism. Then

1. The $g_{\otimes n}$-invariant linear maps $V^{\otimes n}\to\mathbb{R}$ exist only when $n$ is even, and they are spanned by the complete contractions $$x_{\sigma(1)}\otimes\cdots\otimes x_{\sigma(n)}\mapsto g(x_{\tau(1)},x_{\tau(2)})\cdots g(x_{\tau(n-1)},x_{\tau(n)})$$ where $\sigma$ and $\tau$ range over all permutations of $\{1,\ldots,n\}$.
2. If $W\subset V^{\otimes n}$ is a linear subspace such that $(\rho(T))(W)\subset W$ for all $T\in\text{O}(V,g)$ and such that the dimension of the vector space $$\Big\{\text{bilinear symmetric }S:W\times W\to\mathbb{R}\text{ s.t. }(\rho(T))^\ast S=S\quad\forall T\in\text{O}(V,g)\Big\}$$ is one, then there is no nontrivial proper linear subspace $U\subset V$ with $(\rho(T))(U)\subset U$ for all $T\in\text{O}(V,g).$

My reference is chapter 9 in Besse's book on four-dimensional geometry. My questions are the following:

1. Does the above remain true if $g$ is only non-degenerate? It seems like the argument for #2 may break down since it relies on orthogonal projection. I couldn't make a guess on #1 since I'm not familiar with Weyl's book on the classical groups.
2. Where can I find a reference with such a statement? Weyl's book seems to only suggest that the indefinite case is not so different from the definite case, but it's hard for me to tell.

It seems like the article by Peter B. Gilkey "Local invariants of a pseudo-Riemannian manifold", Math. Scand. 36 (1975), 109-130 is relevant, but it's very hard for me to read.

(I asked this question on math.stackexchange with no response.)