  Let $A$ be a finite dimensional algebra over an algebraically closed field
$\mathbf{k}$. If $A$ is quasi-hereditary and the projective dimensions of all
standard modules are at most one, then $A$ is called left strongly
quasi-hereditary. In this paper, we construct a special heredity chain for left
strongly quasi-hereditary algebras. Moreover, we show the quotient algebra by
an ideal which appears in a special heredity chain of left strongly
quasi-hereditary algebra is also left strongly quasi-hereditary algebra.
