C^0({\mathfrak g })&=&{\mathfrak g }& C_0({\mathfrak g })&=&0 \\
C^r({\mathfrak g })&=&[{\mathfrak g },C^{r-1}({\mathfrak g })] & C_r({\mathfrak g })&=&\{x\in {\mathfrak g }:[x,{\mathfrak g }]\in C_{r-1}({\mathfrak g })\}.
\end{array}$$

Note that $C_1({\mathfrak g })$ is by definition the center of ${\mathfrak g }$, which will be denoted by ${\mathfrak z }({\mathfrak g })$.

A Lie algebra ${\mathfrak g }$ is called *k-step nilpotent* if $C^k({\mathfrak g })=\{0\}$ but $C^{k-1}({\mathfrak g })\neq \{0\}$ and clearly $C^{k-1}({\mathfrak g })\subseteq {\mathfrak z }({\mathfrak g })$.

Heisenberg algebras. Let $X_1, \hdots, X_n, Y_1, \hdots, Y_n$ denote a basis of the $2n$-dimensional real vector space $V$ and let $Z\notin V$. Define $[X_i, Y_j]=\delta_{ij} Z$ and $[Z, U]=0$ for all $U\in V$. Thus $\hh_n= V \oplus \RR Z$ is the Heisenberg Lie algebra of dimension $2n+1$, which is 2-step nilpotent.

We shall make use of the notation ${\mathfrak g }'=[{\mathfrak g }, {\mathfrak g }]$ and ${\mathfrak g }''=[{\mathfrak g }',{\mathfrak g }']$. A Lie algebra is called [*2-step solvable*]{} if its commutator is abelian, that is ${\mathfrak g }''=0$.

Let $\mathfrak f_m$ denotes the free Lie algebra on $m$ generators, with $m\geq 2$. (Notice that a unique element spans an abelian Lie algebra). Thus

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