We report optical observations of the Luminous Blue Variable (LBV) HR Carinae which show that the star has reached a visual minimum phase in 2009. More importantly, we detected absorptions due to Si IV 4088-4116 Angstroms. To match their observed line profiles from 2009 May, a high rotational velocity of vrot=150 +- 20 km/s is needed (assuming an inclination angle of 30 degrees), implying that HR Car rotates at ~0.88 +- 0.2 of its critical velocity for break-up (vcrit). Our results suggest that fast rotation is typical in all strong-variable, bona-fide galactic LBVs, which present S Dor-type variability. Strong-variable LBVs are located in a well-defined region of the HR diagram during visual minimum (the LBV minimum instability strip). We suggest this region corresponds to where vcrit is reached. To the left of this strip, a forbidden zone with vrot/vcrit>1 is present, explaining why no LBVs are detected in this zone. Since dormant/ex LBVs like P Cygni and HD 168625 have low vrot, we propose that LBVs can be separated in two groups: fast-rotating, strong-variable stars showing S-Dor cycles (such as AG Car and HR Car) and slow-rotating stars with much less variability (such as P Cygni and HD 168625). We speculate that SN progenitors which had S-Dor cycles before exploding (such as in SN 2001ig, SN 2003bg, and SN 2005gj) could have been fast rotators. We suggest that the potential difficulty of fast-rotating Galactic LBVs to lose angular momentum is an additional evidence that such stars could explode during the LBV phase.
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