Lie algebras with few centralizer dimensions. (Q1867312)

Inspired by results in group theory, the authors consider the structure of complex Lie algebras \(L\) which have the property that centralizers of the elements of \(L\) have only two possible dimensions. Nonnilpotent \(L\) have dim\((L/Z(L))\) less than or equal to three and \(L/Z(L)\) is isomorphic to 1. the two dimensional nonabelian Lie algebra or 2. the three dimensional simple Lie algebra or 3. the three dimensional Lie algebra with basis \({a,x,y}\) and multiplication \([a,x]=x, [a,y]=-y\) and \([x,y]=0.\) If \(L\) is nilpotent, then the nilpotency class of \(L\) is at most 3.