What was Harold Saxon's involvement in the Lazarus experiment?

Question

In series 3 episode 6, The Lazarus Experiment, a connection between Lazarus and Harold Saxon is hinted at by the presence of a shady Saxon agent (played by Bertie Carvel, now better known for his role as Jonathan Strange) who tells Martha's mother certain things about the Doctor.

Later on, in episode 10, The Sound of Drums, Harold Saxon (aka the Master) tells the Doctor:

Professor Lazarus. Remember him and his genetic manipulation device? What, did you think that little Tish got that job merely by coincidence? I've been laying traps for you all this time.

Do we know any more about what Saxon's involvement actually was?

Did he give Lazarus the technology he needed, making the Doctor's claim "No, for once it's strictly human in origin" in the Lazarus episode incorrect? Or was he simply watching, gathering what he needed for his opportunity to strike at the Doctor?

Answer 1

Capital investment

The only clue I can gather from the episode itself is the following line:

Lady Thaw: The people in that room will represent billions of pounds worth of potential investment. Mister Saxon wants to be sure they like what they see.

Lazarus: Don't worry. Our friend will get his money's worth.

^{Doctor Who Series 3 Episode 6: "The Lazarus Experiment"}

This line suggests a pretty mundane involvement on Saxon's part: cash. Whether some of his personal fortune, or in the form of national research grants¹, it's clear that Saxon and/or the British government has some financial stake in Lazarus' research.

According to the BBC, the UK government budgeted £5 billion for science investment in 2007 (the year the episode aired, and presumably roughly when it took place). Precisely how much went to Lazarus is unknown, but evidently a substantial part of Lazarus' funding if Saxon had enough sway to force a hire.

If we grant that Lady Thaw is a representative of Saxon, a later line may hint at a personal financial stake:

Lady Thaw: It's me who made this all possible. This is my triumph, and I will not be denied, not by you, not after everything I've done.

Lazarus: You backed me because you saw a profit. Your concern was financial.

[...]

Lady Thaw: I'll protect my involvement in the project. I'm sure Mister Saxon will be interested... [cut off by Lazarus' imminent transformation into a badly-CGIed scorpion-monster]

^{Doctor Who Series 3 Episode 6: "The Lazarus Experiment"}

Though it's not clear whether this is meant to imply that Thaw was working on behalf of Saxon, or if she invested in the project independently of him.

Scientific Knowledge

In the episode, as mentioned in the question, the Doctor confirms that Lazarus' technology is human:

Martha: I still don't understand where that thing came from. Is it alien?

Doctor: No, for once it's strictly human in origin.

^{Doctor Who Series 3 Episode 6: "The Lazarus Experiment"}

However, this doesn't necessarily preclude the Master's direct involvement, for a few reasons:

1. Martha and the Doctor aren't actually talking about the technology, they're talking about the thing Lazarus has changed into. This is clear from the next lines:

   Martha: Human? How can it be human?

   Doctor: Probably from dormant genes in Lazarus's DNA. The energy field in this thing must have reactivated them. And it looks like they're becoming dominant.

   ^{Doctor Who Series 3 Episode 6: "The Lazarus Experiment"}

   Really all the Doctor is telling us is that Lazarus' machine isn't running off the back of an alien parasite. That doesn't give us any clues about the origin of the machine itself, leaving it open to the possibility that Lazarus is using alien tech.

   Then again, presumably the Doctor would have noticed if there was Gallifreyan technology powering the machine, and would have mentioned something, which leads to my next point.

2. The Master is a genius. He needn't have given Lazarus any extraterrestrial technology at all; it doesn't seem at all implausible for him to have worked out how to make Lazarus' machine work with human technology, and then have given Lazarus some hints to push him in the right direction.

   As Richard points out in comments, Time Lords are demonstrably able to manipulate humans with ease², so it would have been trivial for the Master to lead Lazarus down the right path, all the while having him thinking it was his idea.

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¹ As I am not a British national, and don't know a whole lot about how government-funded scientific research works in the UK (or how it works anywhere else, for that matter), I would deeply appreciate any residents willing to elaborate on the details

² Richard's example is the Doctor's orchestration of a public drop of confidence in Harriet Jones, which he accomplishes with six words.