Is the space $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/\mathrm{SU}(2)$ a symmetric space?

Question

My question is: is the homogeneous space $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/\mathrm{SU}(2)$ a symmetric space?

Definitions: Let $G$ be a connected Lie group and $H$ a closed subgroup. The pair $(G,H)$ is called a symmetric space if there exists an involutive analytic automorphism $\sigma$ of $G$ such that $(H_\sigma)_0\subset H\subset H_\sigma$, where $H_{\sigma}$ is the set of fixed points of $\sigma$ and $( H_\sigma)_0$ is the identity component of $H_{\sigma}.$

Using the notation from wikipedia https://en.wikipedia.org/wiki/Symplectic_group , $\mathrm{Sp}(4,\mathbb{R})$ is a connected Lie group with fundamental group $\mathbb{Z}$ and maximal compact subgroup isomorphic to $\mathrm{U}(2)$. Let $\widetilde{\mathrm{Sp}(4,\mathbb{R})}$ be the universal cover. I observed that the inverse image of $\mathrm{U}(2)$ in $\widetilde{\mathrm{Sp}(4,\mathbb{R})}$ is non-compact, and now the maximal compact subgroup in $\widetilde{\mathrm{Sp}(4,\mathbb{R})}$ becomes $\mathrm{SU}(2)$.

Known information: $\mathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)$ is known to be symmetric, see Cartan's list in Differential Geometry and Symmetric Spaces by Sigurdur Helgason for example. (Different notation! )

I infer that the space might be symmetric, based on the following information:

1. "The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type, $G$ is such a group and $K$ is its maximal compact subgroup." https://en.wikipedia.org/wiki/Symmetric_space As $\widetilde{\mathrm{Sp}(4,\mathbb{R})}$ is simply connected, it follows that $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/SU(2)$ is a symmetric space of non-compact type and of class A there.

2. The answer by YCor Symmetric Spaces via Lie Group Mod Maximal Compact Subgroup "Given a Lie group $G$ and a maximal compact subgroup $K$, when does the quotient $G/K$ admit a Riemannian metric turning it into a symmetric space? I don't have a clear-cut answer. If $G$ is semisimple, every $G$-invariant Riemannian metric on $G/K$ is symmetric. " Observe that the Lie algebra of $\widetilde{\mathrm{Sp}(4,\mathbb{R})}$ is a real simple Lie algebra and hence semisimple. The same result follows.

However I cannot find this space throughout the book Differential Geometry and Symmetric Spaces by Sigurdur Helgason on the detailed classification of symmetric spaces. The other space $\mathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)$ always appears whenever $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/\mathrm{SU}(2)$ should appear.

(Thanks in advance. )

Edit: To clarify the definitions, for the definition of a symmetric space, see the definition of a "symmetric pair" on page 174, Chap IV of the book Differential Geometry and Symmetric Spaces by Sigurdur Helgason. For the definition of a Riemannian symmetric space, see the definition of a "Riemannian globally symmetric space" on page 170 of the book. Both are mentioned in Wikipedia, and I am using the same term as Wikipedia. The Proposition 3.4 on page 174 explains their relations.

Answer 1

As you note both point 1 and 2 explicitly say that $\widetilde{\mathrm{Sp}(4,\mathbb{R})}/SU(2)$, as the quotient of a real semisimple Lie group by a maximal compact subgroup, must be Riemannian symmetric. As to why it doesn't appear in Helgason, it is simply that the classification in Helgason is only up to covers (I assume we are looking at table V on page 518). What you have there is simply a cover of $\mathrm{Sp}(4,\mathbb{R})/\mathrm{U}(2)$ in the list.