Why isn't $SO(n)/SO(n\!-\!1)$ a symmetric space? It's my understanding that one way to define a symmetric space $G/H$ is by the commutation relations
$$ [T^a, T^b] = f^{abc} T^c, \qquad [T^a, X^{\hat{b}}] = f^{a\hat{b}\hat{c}}X^{\hat{c}}, \qquad [X^{\hat{a}}, X^{\hat{b}}] = f^{\hat{a}\hat{b}c} T^c$$
where the generators $\{T,X\}$ of $G$, and the generators $\{T\}$ of $H$ live in the Lie algebra
$$ \mathfrak{h} = \{T\}, \qquad \mathfrak{g} = \{T\} \oplus \{X\}$$
If this is enough to define a symmetric space (such that it then has all the other properties: automorphism $\sigma$, where $\sigma^2 = I$ and so on), then $SO(n)/SO(n-1)$ should be a symmetric space. Its generators satisfy this relation, since the broken generators can be given by
$$\left(\begin{matrix}
0 & \vec{v}\\
-\vec{v}^T & 0
\end{matrix}\right),$$
which clearly commute to give the $T$ generators.
However, it is always said that the symmetric space is $SO(n)/(SO(n-1)\times SO(1))$. Which step am I missing here?
UPDATE: Perhaps it is a symmetric space, as suggested by this master's thesis (p. 14). However other works (p. 3) emphasise that the subgroup should be $SO(n-1) \times SO(1)$. The distinction is vital, because being able to treat $SO(n)/SO(n-1)$ as a symmetric space allows great simplification of symmetry breaking phenomena.
 A: Both spaces: $SO(n)/SO(n-1)$ and also $SO(n)/S(O(n-1) \times O(1)))$ (Please notice the difference from the form given in the question) are symmetric spaces. 
These spaces are locally isomorphic as homogeneous spaces, because the additional group:
$$O(1) \cong \mathbb{Z}_2$$
is discrete. (Due to the fact that it is the group of orthogonal one dimensional matrices, thus it is the discrete group $\{ \pm 1\}$ ). (The notation $S(O(n-1) \times O(1))$, means that we pick up only matrices with unit determinant from the direct product.
Due to the local isomorphism, The Lie algebra decomposition for both spaces is identical and thus both are symmetric spaces:
The  decomposition commutation relation in the form given in the question is sufficient for a homogeneous space to be symmetric.
These spaces are very known symmetric spaces as, the first is a sphere
$$S^{n-1} = SO(n)/SO(n-1)$$
and the second is a real projective space:
$$\mathbb{R}P^{n-1} = SO(n)/S(O(n-1) \times O(1)))$$
(Which is a sphere with each two antipodal points identified).
A basic reference for symmetric spaces is Helgason's book. 
More elementary expositions can be found in the physics literature, for example Appendix: A in McMullan and TsuTsui, and the article by: ArvaniToyeorgos, where the two cases in the question are given in Exapmples 7.
A more advanced reference in which quatum dynamics on homogeneous and symmetric spaces is described, is the report by Camporesi.
