This paper also contains further conditions which are equivalent to being strongly quasi-hereditary, see Theorem 2.
Also, in this paper, as well as in Dlab-Ringel  it is shown that starting with any species without loops, one may construct algebras which are both left and right strongly quasi-hereditary with this species, namely the "deep algebras" () and the "peaked algebras" ().
On Aug 5, 2015, I obtained a message from R. Bautista, asking me to add a note in order to clarify the cooperation of the authors.
Remark: The statement "About Teamwork" starts with the following formulation:
Teamwork requires the observation of a minimal amount of rules: a) Statements or proofs within a team-publication, which are not explicitly attributed to a specified member, belong to the whole team; b) Concerning the internal team-work, the members are bound to an amount of discretion avoiding cacophony.
But the dispute between Gabriel and me concerns the time before the cooperation between Gabriel on the one hand and Bautista, Roiter and Salmeron on the other hand, had started.
The controversy concerns the fact that in [BGRS] the main result, the existence of a multiplicative basis, is contributed to Roiter, whereas it was known that Roiter did not have a complete proof.
I myself refer to this assertion as "theorem of Roiter and Bautista", since I was present when Bautista presented his proof at UNAM, spring 1983. This proof completed the arguments of Roiter (Bautista acknowledged contributions by Salmeron, thus one may question whether one should also refer to him). It is decisive to stress that this presentation was before the cooperation with Gabriel even had started.