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I am trying to test if $a_t$ granger causes $b_t$:

$$a_t=\rho_1+\phi_{11}a_{t-1}+\alpha_{11}b_{t-1}+\beta_{11}c_{t-1}+u_{1t}$$

$$b_t=\rho_2+\phi_{21}a_{t-1}+\alpha_{21}b_{t-1}+\beta_{21}c_{t-1}+u_{2t}$$

$$c_t=\rho_3+\phi_{31}a_{t-1}+\alpha_{31}b_{t-1}+\phi_{31}c_{t-1}+u_{3t}$$

I am trying to use an F test.

However, should be null be: $\phi_{21}$ = 0.

Or do I also need to set: $\beta_{21}$ = 0 because a impacts b through c.

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This site https://www.statisticshowto.com/granger-causality/ shows about conducting the F-test for granger-causality. It also talks about alternative Tests.

"If you have a large number of variables and lags, your F-test can lose power.

An alternative would be to run a chi-square test, constructed with likelihood ratio or Wald tests. Although both versions give practically the same result, the F-test is much easier to run."

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