While the fed has printed a lot of money the last decade and the M1 money quantity growth rate has gone up significantly (red), the M3 growth rate (blue) is almost exactly the same as before the financial crisis.
In the euro area, M1 (red) has grown in the same rate as before, while M3 growth (blue) is significantly lower.
Why have M3 not followed the same quantitive expansion as M1?
And can this be an explanation for why we have not seen stronger inflation?
Why have M3 not followed the same quantitive expansion as M1?
Because commercial banks have used injected liquidities to "consolidate" their balance sheet instead of credit-stimulating demands. What I mean by "consolidate" is that the subprime crisis and its consequences have contaminated the Euro-zone's commercial banks through securitization, interrogating the default risk associated to their assets and liabilities. Injected liquidities have thus often been used by commercial banks to anticipate these potential defaults on the one hand, and to elevate their fractional-reserve ratios on the other hand so as to comply with the Basel III framework.
And can this be an explanation for why we have not seen stronger inflation?
Precisely. These precautious behaviours led the money multiplier to be somehow blocked (to ~$1$), i.e. the expansion of M3 is the main conventional inflation engine.
As explained by Kent (with who I fully agree as you can see), on March 10, 2016, the ECB has even considered the possibility of giving 200€ per month and per person to stimulate inflation. And this with the intention of circumventing the commercial-bank system. This unconventional monetary policy is called monetary helicopter. And actually, as of September 2019, the ECB is still considering the possibility of using this non-conventional monetary instrument.
As you point out, central banks have "printed" a considerable amount of money since the 2008 crisis. The following chart of the US monetary base is one of clearest examples of this phenomenon.
The metric you seem to be pointing out through your question is known as the money multiplier: the ratio of broad money (e.g., M1, M2, or MZM) to the monetary base. As we can see with the following chart, in the US, the money multiplier has decreased since the 2008 crisis.
The money multiplier grows when banks and other financial institutions make loans. So in short, M3/MZM hasn't grown because a significant amount of the monetary base has just been kept in bank reserves without being loaned out.
This has been used as an argument on the ineffectiveness of quantitative easing, and that we need to more directly put money into people's hands in order to stimulate the economy.
And if excess reserves were all loaned out and invested/spent, we should indeed see higher levels of inflation in the short term. If the money is invested/spent wisely, then productivity growth may offset that inflation on a longer term assuming we don't continue to increase the money supply.
