While preparing a seminar I gave today, the following question arose. I asked the seminar participants, but nobody knew the answer. Hence I'm asking it here in MO.

**Background**

Recall that a complete, connected and simply connected pseudoriemannian manifold $(M,g)$ is a *symmetric space* if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection: $\nabla R = 0$.

Typical examples are the (simply-connected) spaces of constant curvature: sphere, hyperbolic space, (anti) de Sitter spacetimes,... all of which admit local isometric embeddings as quadrics in some flat pseudoriemannian manifold $\mathbb{R}^{p,q}$. Recall that the flat metric on $\mathbb{R}^{p,q}$ is given in flat coordinates by $$\sum_{i=1}^p (dx_i)^2 - \sum_{i=1}^q (dx_{p+i})^2.$$

For example, the sphere with unit radius of curvature embeds in $\mathbb{R}^{n+1}$ as the quadric $$x_0^2 + x_1^2 + \cdots + x_n^2 = 1,$$ whereas the hyperbolic space embeds in $\mathbb{R}^{1,n}$ as one sheet of the quadric $$-x_0^2 + x_1^2 + \cdots + x_n^2 = -1,$$ again for unit radius of curvature.

Similarly, and again for unit radii of curvature, $n$-dimensional de Sitter spacetime is the universal covering space of the quadric $$-x_0^2 + x_1^2 + \cdots + x_n^2 = 1$$ in $\mathbb{R}^{1,n}$, whereas $n$-dimensional anti de Sitter spacetime is the universal covering space of the quadric $$-x_0^2 + x_1^2 + \cdots + x_{n-1}^2 - x_n^2 = -1.$$

This continues to be the case for other spaces of constant curvature in other signatures.

Other riemannian symmetric spaces, such as the grassmannians, can also admit isometric embeddings, this time in projective spaces, whose image is the intersection of a number of quadrics. This is the celebrated Plücker embedding. Notice that grassmannians do not (generally) have constant sectional curvature.

The remaining nontrivial lorentzian symmetric spaces -- the $n$-dimensional Cahen-Wallach spaces -- can also be locally embedded isometrically in $\mathbb{R}^{2,n}$ as the intersection of two quadrics. In particular this shows that all the indecomposable lorentzian symmetric spaces (in dimension $>1$, at least), which are the (anti) de Sitter and Cahen--Wallach spacetimes, can be locally embedded isometrically as the intersection of quadrics in some pseudoeuclidean space.

**Question**

Is this also the case for the other simply-connected (pseudo)riemannian symmetric spaces?

Perhaps asking about quadrics is too strong, so perhaps a weaker question is

Are simply-connected symmetric spaces always (locally) algebraic?

Here by locally algebraic I mean that they are the universal covering space of an algebraic space.