On Nichols (braided) Lie algebras

We prove {\rm (i)} Nichols algebra $\mathfrak{B}(V)$ of vector space $V$ is finite-dimensional if and only if Nichols braided Lie algebra $\mathfrak{L}(V)$ is finite-dimensional; {\rm (ii)} If the rank of connected $V$ is $2$ and $\mathfrak{B}(V)$ is an arithmetic root system, then $\mathfrak B(V) = F \oplus \mathfrak L(V);$and(\mathrm{i}\mathrm{i}\mathrm{i})if$\Delta (\mathfrak B(V))$ is an arithmetic root system and there does not exist any $m$-infinity element with $p_{uu} \not= 1$forany$u \in D(V)$,then$\dim (\mathfrak B(V) ) = \infty$ if and only if there exists $V^{\mathrm{\prime }}$, which is twisting equivalent to $V$, such that $\dim ({\mathfrak{L}}^{-}(V^{\mathrm{\prime }}))=\mathrm{\infty }\mathrm{.}$ Furthermore we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.