Since the Supreme Court opened for business three years ago, with its premises outside the precincts of Parliament, there has been a widespread impression that the UK now had a system with complete separation of powers, similar to that in the United States. In fact, though, the links between the legislature and the executive, on the one hand, and the judiciary, on the other, were not entirely severed.

As we have been reminded this week, the Justice Secretary and Lord Chancellor – the job currently held by Chris Grayling – retains the power to appoint the court's chief executive and other administrative officers. And as we also learnt, this is not to the judges' liking.

In a proposed amendment to the Crime and Courts Bill, which had its third reading in the House of Lords yesterday, the former president of the Supreme Court, Lord Phillips, is trying to change this. He wants responsibility for these appointments to be transferred to the court's sitting president, a position that his successor, Lord Neuberger, supports. The immediate practical argument is that the judges on the spot appreciate the court's needs better than ministers, whose priorities might be different.

A further argument is that the chief executive would then owe his or her loyalty unambiguously to the court, rather than to the government of the day. And the conclusion must be that, so long as the current system operates, the appearance of Supreme Court independence does not fully match the reality.

Lord Phillips, who quite properly waited until he retired from the court to press for this change, should be heeded. The creation of the Supreme Court represented a decisive move towards the separation of constitutional powers, and that process should be completed. Having avoided the chief downside of the US system – the highly politicised nature of the Supreme Court justices, whose appointment is in the gift of the President – the UK's highest court should be allowed to capitalise on its success so far and break this last government link.