  The $KBc$ algebra is a subalgebra that has been used to construct classical
solutions in Witten's open string field theory, such as the tachyon vacuum
solution. The main purpose of this paper is to give various operator sets that
satisfy the $KBc$ algebra. In addition, since those sets can contain matter
operators arbitrarily, we can reproduce the KOS and the Erler-Maccaferri
solutions. Starting with a single D-brane solution on the tachyon vacuum, we
replace the original $KBc$ in it with an appropriate set to generate each of
the above solutions. Thus, it is expected that the $KBc$ algebra, combined with
the single D-brane solution, leads to a more unified description of classical
solutions.
