  Let $B$ be a representation-finite $\mathbb{C}$-algebra.
  The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by
Ch. Riedtmann.
  If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra
associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove
that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any
representation-directed $\mathbb{C}$-algebra $B$.
