In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for $mathfrak L(V)${tiny $_{R}$} and $mathfrak L(V)${tiny $_{L}$} over (i) the quantum linear space $V$ with $dim V=2$; (ii) a connected braided vector $V$ of diagonal type with $dim V=2$ and $p_{1,1}=p_{2,2}= -1$. We give the sufficient and necessary conditions for $mathfrak L(V)${tiny $_{R}$}$= mathfrak L(V)$, $mathfrak L(V)${tiny $_{L}$}$= mathfrak L(V)$, $mathfrak B(V) = Foplus mathfrak L(V)${tiny $_{R}$} and $mathfrak B(V) = Foplus mathfrak L(V)${tiny $_{L}$}, respectively. We show that if $mathfrak B(V)$ is a connected Nichols algebra of diagonal type with $dim V>1$, then $mathfrak B(V)$ is finite-dimensional if and only if $mathfrak L(V)${tiny $_{L}$} is finite-dimensional if and only if $mathfrak L(V)${tiny $_{R}$} is finite-dimensional.
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