A reduced root system $\Delta$ on a finite dimensional Euclidean space $V$ is called a normal $\sigma$-system of roots if $\exists$ a linear involutive isometry $\sigma:V\to V$ such that $\sigma\Delta=\Delta$ and $\sigma \alpha-\alpha\notin \Delta$ $\forall \alpha\in \Delta$.
Let $V=V^{+}\oplus V^{-}$ be the orthogonal decomposition of $V$ into corresponding eigenspaces of $\sigma$.Now my question is the following.
If I look at the orthogonal projection of all elements of $\Delta$ onto $V^{+}$ such that projections are non-zero,then would that collection of projections also form a root system (not necessarily reduced) inside $V^+$?
If the answer is positive then is there any quick way to verify that?
Relevance of this question: I have been studying the classification of real simple lie algebras where understanding some specific kind of involution(like mentioned in the question) of the root system of the complexified lie algebra becomes important.
Thanks in advance.
