A jury rejected the defendant’s claim that he lacked criminal responsibility and convicted him of murder in the first degree based on extreme atrocity or cruelty.1 At trial the defendant did not dispute that he committed the murder, but claimed that he was not criminally responsible. The defendant appeals from his conviction claiming that (1) the trial judge erred by admitting evidence of the defendant’s alleged prior bad acts and (2) the judge’s instructions on criminal responsibility were erroneous. He also requests that we direct the entry of a verdict of not guilty by reason of insanity, reduce the verdict to manslaughter, or grant a new trial pursuant to our power under G. L. c. 278, § 33E. We affirm the conviction and decline to exercise our power under G. L. c. 278, § 33E.
*125Facts. We recite the facts in the light most favorable to the Commonwealth. On the night of August 28, 1995, the defendant killed his wife by beating her with a rock. He ruptured her right eye and the surrounding bones, and destroyed her face beyond recognition; he sliced her torso, removed her organs, impaled them on a stake and left the stake lying in the backyard of their home. He then drove about aimlessly with their four and one-half month old baby in the back seat. He later claimed that he believed his wife was an “enemy alien vampire,” part of an invasion.
The defendant and the victim met at John Hancock Mutual Life Insurance Company (John Hancock) where they both were employed. They married in September, 1991, but their relationship was less than idyllic. A coworker testified that in the summer of 1990, before the couple were married, and approximately five years before the murder, the victim appeared at work with a blackened left eye and a bruise on her left cheek. The coworker asked the defendant what happened and he stated that the victim had “walked into a door.” When the coworker repeated the question, the defendant admitted that he and the victim had an argument and he “may have pushed her” and she had fallen and hit her face. Three other coworkers testified that in September, 1993, approximately two years before the murder, the victim had a black right eye. One of the coworkers testified that the victim tried to cover the bruise with makeup and avoided showing her face. In response to an inquiry from another co worker, the defendant said, “[The victim] walked into a door or fell down.”2
On February 18, 1994, the victim delivered a baby boy, prematurely. The baby died after only a few hours due to an accumulation of fluid around the heart and lungs, a condition that often causes heart failure. The defendant was extremely depressed by this loss and became obsessed with the baby’s death. He generally ignored their daughter who was born in April, 1995.
In March, 1995, the defendant told a co worker that everything was “so difficult and hard now”; the coworker told the defendant that she interpreted this statement as meaning that he was having problems with his wife and the defendant did not state otherwise. There was tension between the couple during *126the spring of 1995 because the defendant was not paying enough attention to their baby daughter. In June, 1995, the victim planned to return to work and the defendant was indifferent to her distress about putting their baby daughter into daycare.3 He did not console her when she was crying about this plan. The defendant controlled the money the victim spent to the extent that she had to ask him for money for the day before he left for work or else had to call him at work for approval to spend money. She could not adjust the thermostat while he was at home without his permission. During the summer of 1995, the tension between the couple grew because the defendant was not attentive to their new daughter, but was concentrating instead on their dead son. The victim became “very upset” with the defendant in July when she learned that, while she was not at home, he had strapped their baby to the changing table and left her unattended in the bathroom while he was in the kitchen grinding coffee beans. In August, the defendant wanted to have another child despite the fact that another pregnancy might have required an invasive, painful procedure and his wife was uncertain about the prospect. In mid-August, when the victim brought the baby to lunch with the defendant in the John Hancock cafeteria, the defendant took the baby to another coworker against the victim’s wishes and the victim became angry.
At John Hancock, the defendant was the director of acquisitions and disposal, earning $96,000 per year plus bonuses. His work required him to engage in high-level complex tasks in the weeks before and on the day of the murder.
1. Prior bad acts evidence. The defendant claims that the Commonwealth should not have been permitted to introduce evidence that the victim had sustained two black eyes in the years before the killing for several reasons: it was not established that he had inflicted those injuries on the victim; the injuries were too remote to be relevant; and the potential for prejudice from such evidence outweighed any probative value. We reject his contentions.
“Before prior bad act evidence can be admitted against a defendant, the Commonwealth must satisfy the judge that ‘the jury [could] reasonably conclude that the act occurred and that the defendant was the actor.’ Huddleston v. United States, 485 U.S. 681, 689 (1988). The Commonwealth need only show *127these facts by a preponderance of the evidence. See Care & Protection of Laura, 414 Mass. 788, 792 (1993), and cases cited; Commonwealth v. Robinson, 146 Mass. 571, 581-583 (1888); P.J. Liacos, Massachusetts Evidence § 4.4.8, at 170 (6th ed. 1994 & Supp. 1995). If the judge finds this standard has been met, it is thereafter for the jury to evaluate the evidence for ‘its weight and credit.’ Commonwealth v. Robinson, supra at 582.” Commonwealth v. Leonard, 428 Mass. 782, 785-786 (1999).4
The evidence indicated that the defendant had lied about the 1990 incident and that he admitted he may have been the cause of that black eye. Three years later, the defendant offered a similar reason as the cause of a second black eye. The injury was an unusual one for a young woman to suffer and the jury could infer that, if the defendant admitted to being the cause of the first injury, he was the cause of the second. This inference is strengthened by the fact that the reason given both times was highly suspicious. When one falls or walks into a door, a black eye is not a likely result. The nose or forehead hits the object first and protects the eye socket. Further, the victim was clearly ashamed of the fact that she had a black eye. An accidental injury as a result of a fall does not ordinarily cause shame, but an injury from an abusive husband does. Thus, it is reasonable to infer that the defendant’s statement was a fabrication and that he was the source of both injuries.
The evidence regarding the black eyes was relevant to the issues of motive and intent and to the defendant’s claim of lack of criminal responsibility. “Evidence of a hostile relationship between a defendant and his spouse may be admitted as relevant to the defendant’s motive to kill the victim spouse.” Commonwealth v. Gil, 393 Mass. 204, 215 (1984). Further, evidence of hostility close in time to the murder renders relevant earlier evidence of physical abuse of the victim by the spouse. Id. at 217 (evidence of physical abuse by husband against wife four and five years before murder became relevant “by virtue of” *128evidence of continuing hostility between them during a time much closer to the killing). Here, as in the Gil case, there was evidence of hostility during the months preceding the murder: the defendant’s statement in March, 1995, that everything was “so difficult and hard now” (interpreted as a statement regarding his marital relationship); the defendant’s lack of concern in June for his wife’s distress about care for their baby while she worked; the victim’s annoyance at the defendant for not attending to their daughter; his determination in August to have another child despite his awareness that this might be very painful for his wife and that she was uncertain whether to have another child; and the victim’s anger at the defendant in August when she visited the defendant for lunch in the John Hancock building. Thus, the testimony concerning the wife’s black eyes “became relevant by virtue of the evidence of a continuing hostility between the [Rosenthals] during a time in much closer proximity to the killings.” Id.
The evidence regarding the black eyes was also relevant to establish the defendant’s state of mind, thereby rebutting his claim that he lacked criminal responsibility when he committed the murder. See Commonwealth v. Ashman, 430 Mass. 736, 741 (2000) (where defendant claimed mental illness or impairment, evidence of prior altercation between defendant and victim one month before murder admissible to show defendant’s state of mind, intent, and relationship with victim). Evidence of a hostile relationship and an intentional act grounded in that hostile relationship is clearly probative as to the defendant’s claim that he was not criminally responsible at the time of the murder. The infliction of the black eyes which might otherwise have been remote in time thus become relevant as part of a continuum of hostile behavior.
Although ordinarily the evidence would be remote, the uniqueness of the injury and the striking similarity between the prior injuries and those inflicted during the murder establish the relevance of the earlier black eyes. When he killed his wife, the defendant totally ruptured her right eye and the bones around it; this is a more severe version of the previous black eyes. See Commonwealth v. Snell, 428 Mass. 766, 777, cert. denied, 527 U.S. 1010 (1999) (victim told neighbor that defendant had tried to smother her eighteen months before death by asphyxia due to smothering). This distinctive type of injury renders the earlier acts, otherwise remote, relevant.
*129As the evidence was probative of the defendant’s relationship with the victim and his state of mind; relevant to his claim of lack of criminal responsibility; and not too remote, the judge acted within his sound discretion in determining that it was admissible and that its probative value outweighed its prejudicial effect. The judge also gave appropriate limiting instructions to the jury concerning this evidence both when the evidence was admitted and in his final instructions to the jury.5
2. Instruction on extreme atrocity or cruelty. The defendant further contends that the judge erred by not instructing the jury correctly concerning their use of evidence of mental impairment in determining whether the murder had been committed with extreme atrocity or cruelty. The defendant did not object to the instruction on this ground. We consider it to determine whether the instruction created a substantial likelihood of a miscarriage of justice. Commonwealth v. Wright, 411 Mass. 678, 681 (1992). An instruction on extreme atrocity or cruelty requires a listing of the Cunneen factors. See Commonwealth v. Cunneen, 389 Mass. 216, 227 (1983) (jury must base their verdict of murder in first degree committed with extreme atrocity or cruelty on one of listed factors).6 The defendant does not claim that the judge failed to instruct regarding the Cunneen factors. He argues, rather, that pursuant to Commonwealth v. Gould, 380 Mass. 672, 686 n.16 (1980), the jury must be instructed to consider the defendant’s ability to appreciate the extreme atrocity or cruelty of his acts before they may consider any of the Cunneen factors. He contends that, except for those Cunneen factors relating to the defendant’s indifference to or pleasure in the victim’s suffering, the factors concern the victim’s state of mind or the physical acts themselves and the instructions do not explain that the jurors are to consider the defendant’s mental state on the other factors, such as the ones concerning the number, manner, and force of the blows. Therefore, he argues, *130the judge was required to instruct that the jury should consider the defendant’s mental state on each individual Cunneen factor.
Pursuant to Commonwealth v. Gould, supra, evidence of the defendant’s mental illness is to be considered on the issue of whether "a murder was committed with extreme atrocity or cruelty. The judge here directed the jurors to consider each of the Cunneen factors and he Usted them accurately. He instructed the jurors further that they must consider all the Cunneen factors and base any verdict on this theory on evidence of at least one of the listed factors. He instructed on the defendant’s mental impairment in the required manner:
“You may consider the defendant’s mental condition at the time of the kilhng, including the effect of any mental impairment, if any, on all of the factors that are relevant in determining whether or not the Commonwealth has proven beyond a reasonable doubt that the defendant committed the murder with extreme atrocity or cruelty.”7
The judge properly instructed the jury as to mental impairment and its effect on extreme atrocity or cruelty. Our case law does not require that the jury be instructed as the defendant requests, and we see no reason to alter the instructions.
3. Relief pursuant to G. L. c. 278, § 33E. The defendant asks us to exercise our extraordinary power and enter a verdict of not guilty by reason of insanity, reduce the verdict to manslaughter, or grant a new trial pursuant to G. L. c. 278, § 33E. We are not aware of any case in which we have ordered the entry of a judgment of not guilty by reason of insanity after a jury have rejected such a defense. Commonwealth v. Keita, 429 Mass. 843, 848 (1999). We have considered the entire record pursuant to our obligation under G. L. c. 278, § 33E. We are not prepared to second-guess the jury’s verdict or to reduce the verdict to manslaughter or to order a new trial. The defendant received a *131fair trial and the jury’s verdict is consistent with the evidence that the defendant was criminally responsible when he killed his wife and that the killing was committed with extreme atrocity or cruelty.
Judgment affirmed.
The defendant was also tried on a theory of deliberate premeditation, but the jury did not convict him on this theory.
The victim stopped working when she became pregnant in 1995.
The victim returned to work briefly in July, 1995, but left after two weeks “for family reasons.”
Although the Commonwealth must, of course, prove all essential elements of the crime beyond a reasonable doubt (and, because the issue was raised, that the defendant was criminally responsible), preliminary questions of fact need only be proved by a preponderance of the evidence. See Care & Protection of Laura, 414 Mass. 788, 791-792 (1993). See also Commonwealth v. Azar, 32 Mass. App. Ct. 290, 309 (1992) (evidence tending to show defendant’s prior bad acts and state of mind need not be proved beyond reasonable doubt).
The defendant’s reliance on cases such as Commonwealth v. Magraw, 426 Mass. 589, 593-597 (1998), is misplaced. No hearsay statements of the victim were admitted here.
These factors are the defendant’s indifference to or pleasure in the victim’s suffering, the victim’s consciousness and degree of suffering, the extent of the victim’s physical injuries, the number of blows delivered by the defendant, the manner and force with which the defendant delivered the blows, the weapon or weapons used by the defendant, and the disproportion between the means needed to cause death and the means used by the defendant. See Commonwealth v. Cunneen, 389 Mass. 216, 227 (1983).
The judge also instructed the jury that they were to consider any mental impairment on the third prong of malice and referred to this instruction when he instructed on the Cunneen factors (Commonwealth v. Cunneen, 389 Mass. 216 [1983]). In addition, the judge gave the Gould mental impairment instruction twice again, in defining deliberate premeditation and murder in the second degree. In response to a question from the jury, the judge repeated the Gould instruction four more times (Commonwealth v. Gould, 380 Mass. 672 [1980]), in relation to deliberate premeditation, malice, extreme atrocity or cruelty, and murder in the second degree.