Structures of Nichols (braided) Lie algebras of diagonal type

Let $V$ be a braided vector space of diagonal type. Let $mathfrak B(V)$, $mathfrak L^-(V)$ and $mathfrak L(V)$ be the Nichols algebra, Nichols Lie algebra and Nichols braided Lie algebra over $V$, respectively. We show that a monomial belongs to $mathfrak L(V)$ if and only if that this monomial is connected. We obtain the basis for $mathfrak L(V)$ of arithmetic root systems and the dimension for $mathfrak L(V)$ of finite Cartan type. We give the sufficient and necessary conditions for $mathfrak B(V) = Foplus mathfrak L^-(V)$ and $mathfrak L^-(V)= mathfrak L(V)$. We obtain an explicit basis of $mathfrak L^ - (V)$ over quantum linear space $V$ with $dim V=2$.