If it wasn't specifically included at clause 4 the offence wouldn't be triable summarily, because the 25 year maximum sentence far exceeds the limit allowed the Magistrates' Court of Victoria by s 28.
Not all types of aggravated burglary can be dealt with summarily, though. Schedule 2 clause 4.7 provides,
4.7 Offences under section 77 of the Crimes Act 1958 (aggravated burglary), if the offence involves an intent to steal property the amount or value of which does not in the judgment of the court exceed $100 000.
A similarly worded provision relating to simple burglary is found just above it at sub-clause 4.6. Consequently, an aggravated burglary where the burglar has an intention to steal may be dealt with summarily, but a burglary where they have the intention to assault or damage cannot: DPP v Vergios  VSC 97.
The two circumstances of aggravation provided in the Act are,
a) at the time has with him or her any firearm or imitation firearm, any offensive weapon or any explosive or imitation explosive; and
(b) at the time of entering the building or the part of the building a person was then present in the building or part of the building and he or she knew that a person was then so present or was reckless as to whether or not a person was then so present.
In DPP v Woodward  VSC 299 a magistrate dismissed a charge of aggravated burglary because, although it was admitted that the accused was carrying a pocket knife with him at the time of the burglary, he considered it did not meet the definition of an offensive weapon. On judicial review Cavanough J found that there was no error on the face of the record, but stressed [at 47] that his finding was not conclusive on the point. There is more judicial consideration of the status of pocket knives in R v Pope  VSCA 108, referring to Wilson v Kuhl; Ryan v Kuhl  VR 315 which involved a carving knife.
It's not necessary for the burglar to have originally acquired the item with the intention of using it offensively, provided such an intention existed at the time of the burglary: R v Nguyen  1 VR 551.
UK authorities (considering legislation similar to our own) favour the view that the time when being armed falls to be considered is at the time of theft rather than the time of entry. An intruder who arms himself with a weapon after entry is guilty of aggravated burglary: R v O'Leary (1986) 82 Cr App R 341. The offender entered the house unarmed but picked up a kitchen knife once inside and used it to force the occupier to hand over property. Similarly, where a burglar brandished a screwdriver he had brought with him when surprised by residents it was considered a weapon, although originally intended as a tool to gain entry: R v Kelly(1993) 97 Cr App R 245.