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In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A=\left[\begin{array}{cc}U& 0\\ M& V\end{array}\right]$, we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and ${}_{V}M$ is a good V-module.