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Heat the oven to 180C (160C fan)/350F/gas mark 4, and grease and line a 20 cm square tin with baking paper, leaving an overhang on the sides.
Put the butter in a small saucepan on a low heat, then, once melted, add the oil and leave to cool slightly.
In a small bowl, sift the flour, baking powder, baking soda and salt.
In a large bowl, rub the sugar and lemon zest with your fingertips, until fragrant and damp.
Pour in the melted butter mix, whisk until smooth and combined, then add the eggs one at a time, whisking well after each addition, until the mixture thickens and turns glossy.
Whisk in the lemon juice, then add the dry ingredients, gently stirring and folding with the whisk until the mix is smooth and fully incorporated.
Scrape the batter into the lined tin, smooth the top and bake for about 35 minutes, or until a skewer inserted into the centre of the cake comes out clean.
Remove from the oven and put the tin on a rack to cool.
To make the glaze, in a small bowl whisk the icing sugar and lemon juice to a thick but pourable consistency.
Spoon this over the top of the cooled cake and spread out into a thin, even layer.
Scatter over the pistachios straight away, before the glaze begins to set, then cut into squares to serve.