Cartan subspace of graded Lie algebras

Suppose $$\mathfrak{g}$$ is a complex reductive Lie algebra and $$\theta$$ is an automorphism of order $$2$$. Let $$\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$$ be the corresponding $$\mathbb Z_2$$-gradation. Cartan defined the concept of Cartan subspace and the Weyl group of the symmetric pair $$(\mathfrak{g}, \mathfrak{g_0})$$. By definition a Cartan subspace of the pair $$(\mathfrak{g}, \mathfrak{g_0})$$ is a subspace $$\mathfrak{c}$$ of $$\mathfrak{g}_1$$ which is a maximal commutative subspace consisting of semisimple elements and the Weyl group is by definition is the quotient $$N_{\mathfrak{g}_0}(\mathfrak{c})/Z_{\mathfrak{g}_0}(\mathfrak{c})$$. What is a Cartan subspace and the Weyl group for the pair $$(\mathfrak{sl}_{2n}, \mathfrak{sp}_{2n})$$ ?. Is there a canonical choice for $$\mathfrak{c}$$ in this case ? What I understood from different sources is that the Weyl group is a subgroup/quotient of $$S_{2n}$$. Can it be described explicetely ?

For a general symmetric space (let’s say over $$\mathbb{C}$$ for simplicity) to obtain a Cartan subspace, take a Cartan subalgebra $$\mathfrak{t}$$ of $$\mathfrak{g}$$ that is stable under the involution $$\theta$$. This decomposes $$\mathfrak{t}=\mathfrak{t}_0\oplus \mathfrak{t}_1$$ in terms of the eigenspaces of $$\theta$$. Choose such a Cartan so that the fixed subspace $$\mathfrak{t}_0$$ is of minimal dimension. The $$-1$$ eigenspace $$\mathfrak{t}_1$$ gives a Cartan subspace of the pair $$(\mathfrak{g},\mathfrak{g}_0)$$. Over $$\mathbb{C}$$, tori with this property (referred to as maximally $$\theta$$-spit in the literature) are characterized as those lying in a Borel subalgebra $$\mathfrak{b}$$ such that $$\theta(\mathfrak{b})$$ is the opposite Borel containing $$\mathfrak{t}$$: that is, $$\mathfrak{b}\cap\theta(\mathfrak{b})=\mathfrak{t}.$$
In the case you mention, take the diagonal Cartan of $$\mathfrak{sl}(2n)$$ and the involution $$\theta(X)=-JX^TJ,$$ where $$J$$ is the standard "opposite diagonal" symplectic form.
As for the Weyl group, a result of Luna shows that if $$W$$ is the Weyl group of $$(\mathfrak{g},\mathfrak{t})$$ with $$\mathfrak{t}=\mathfrak{t}_0\oplus \mathfrak{t}_1$$ as above, consider the two subgroups $$W_1=\{w\in W: w(\mathfrak{t}_1)=\mathfrak{t}_1\}$$ and $$W_2=\{w\in W_1: w|_{\mathfrak{t}_1}\equiv 1\},$$ then $$W(\mathfrak{g}_0,\mathfrak{t}_1)\cong W_1/W_2$$.