= Synagoge alpha4 (Lexica Segueriana 3.14); Photius, Lexicon alpha30 Theodoridis; perhaps ultimately derived in part from Phrynichus (Praeparatio rhetorica fr. 39 de Borries); cf. Hesychius alpha89 and a cluster of related entries: alpha 2276, Hesychius alpha4899, Synagoge alpha589, Photius alpha1845.
[1] ba/sanos can mean both the touchstone itself and the testing process. See beta 139, and cf. beta 137.
[2] Aelian fr.9 Hercher (= 9 Domingo-Forasté). The version of the entry at Synagoge alpha4 includes the information that this is from the third book of the work in question.

The (neuter) headword is the plural of alpha 131 (and cf. alpha 132). It is perhaps, though not necessarily, quoted from somewhere.
[1] Homer, Iliad 4.144 (web address 1), on an ivory cheek-piece for a horse.
[2] This fragment of Hesiod (142 Merkelbach-West, 233 Rzach) is not known from any other source. It may pertain to the story of Europa in the Catalogue of Women.
[3] Pindar, Nemean Odes 10.125 (67 Bowra): web address 2.
[4] Euripides, Alcestis 613: web address 3.
[5] Already at alpha 131.
[6] Used as prizes for victors in competition.
[7] Same material in Photius (Lexicon alpha92 Theodoridis) and elsewhere; cf. Kassel-Austin, PCG II p.365 (on Antiphanes fr.102).

LSJ entry at web address 1; and cf. generally alpha 234.
Same material in other lexica (references at Photius alpha176 Theodoridis), and also in the scholia to Plato, Axiochus 371D, where the headword -- accusative plural, not genitive singular -- occurs.

[1] This meaning is not attested in LSJ (web address 1 below). Perhaps it stems from a misunderstanding of the Aristophanes passage about to be quoted, where in addition to applying the anchor-hold, Kleon is charged with squeezing treasury officials like ripe figs.
[2] Aristophanes, Knights 262-3 (web address 2), with comment from the scholia there.

The opening material here is also in Photius and other lexica.
[1] The second gloss here is a mistake (perhaps by confusion with the following entry, where the same word, a)gkw/n, is translated 'embrace').
[2] An a)/gos is "any matter of religious awe": LSJ s.v.; see also pi 150.
[3] Thucydides 1.126.2-12 (web address 1), here so drastically abridged as to be misleading. (This banishment was part of the events of 632 BCE, now relevant two centuries later in the build-up to the Peloponnesian War. The original 'accursed' had returned -- and nobody was banished in 432.)
[4] From Philoponus, Differences. (For this epic/poetic noun see LSJ s.v.)

Apart from the initial glossing (on which see next note), this material also occurs in Photius, Lexicon alpha316 Theodoridis.
[1] The word used for 'competitions' here is the (accusative) plural of the headword itself. The Suda seems therefore to be saying, indirectly, that the word denotes both competition and the training for it.
[2] i.e. the arena. Adler cites Homer, Iliad 23.273 for this; Theodoridis chooses Odyssey 8.260. For instances in other authors (including the one about to be quoted here) see LSJ s.v. a)gw/n I.2.
[3] Thucydides 5.50.4, on Lichas the Spartan.

[1] Sophocles, Ajax 572-3 (web address 1 below); again at lambda 839.
[2] (Also in the paroemiographers, e.g. Apostolius 1.25.) Possibly Contest, the divine personification of the agon (cf. Pausanias 5.26.3), though the apparently personifying language does not guarantee this. See further, next note.
[3] Used in Plato, Cratylus 421D (where a scholiast cited Aristophanes fr. 321 Kock as an earlier attestation of it) and Laws 751D. Also in the paroemiographers, e.g. Gregorius 1.11.

The headword is future participle, masculine nominative plural, of the verb a)gwni/zomai. It is perhaps extracted from the quotation given, though not demonstrably so, and there are plenty of alternatives (beginning with Thucydides, Xenophon and Plato).
[1] Quotation unidentifiable, but evidently from (or connected with) a war narrative. This brings out the point that the verb in question has military as well as athletic overtones.

[1] This glossing shows that the headword, a neuter noun, is in the accusative case, and thus extracted from somewhere other than the quotation about to be given (where it is nominative).
[2] Quotation (also in the Lexicon Vindobonense) unidentifiable.
[3] Aristophanes, Frogs 283-4 (web address 1 below). For the sense of a)gw/nisma here (both a struggle and its reward) Dover ad loc. compares Thucydides 7.59.2 (web address 2 below).

An interesting distinction, but uncorroborated outside lexicography.